id sid tid token lemma pos american_scientific_journal-3517 1 1 90 90 NUM american_scientific_journal-3517 1 2 american american ADJ american_scientific_journal-3517 1 3 scientific scientific ADJ american_scientific_journal-3517 1 4 research research NOUN american_scientific_journal-3517 1 5 journal journal NOUN american_scientific_journal-3517 1 6 for for ADP american_scientific_journal-3517 1 7 engineering engineering NOUN american_scientific_journal-3517 1 8 , , PUNCT american_scientific_journal-3517 1 9 technology technology NOUN american_scientific_journal-3517 1 10 , , PUNCT american_scientific_journal-3517 1 11 and and CCONJ american_scientific_journal-3517 1 12 sciences science NOUN american_scientific_journal-3517 1 13 ( ( PUNCT american_scientific_journal-3517 1 14 asrjets asrjet NOUN american_scientific_journal-3517 1 15 ) ) PUNCT american_scientific_journal-3517 1 16 issn issn PROPN american_scientific_journal-3517 1 17 ( ( PUNCT american_scientific_journal-3517 1 18 print print NOUN american_scientific_journal-3517 1 19 ) ) PUNCT american_scientific_journal-3517 1 20 2313 2313 NUM american_scientific_journal-3517 1 21 - - SYM american_scientific_journal-3517 1 22 4410 4410 NUM american_scientific_journal-3517 1 23 , , PUNCT american_scientific_journal-3517 1 24 issn issn PROPN american_scientific_journal-3517 1 25 ( ( PUNCT american_scientific_journal-3517 1 26 online online ADJ american_scientific_journal-3517 1 27 ) ) PUNCT american_scientific_journal-3517 1 28 2313 2313 NUM american_scientific_journal-3517 1 29 - - SYM american_scientific_journal-3517 1 30 4402 4402 NUM american_scientific_journal-3517 1 31 Β© Β© PROPN american_scientific_journal-3517 1 32 global global ADJ american_scientific_journal-3517 1 33 society society NOUN american_scientific_journal-3517 1 34 of of ADP american_scientific_journal-3517 1 35 scientific scientific ADJ american_scientific_journal-3517 1 36 research research NOUN american_scientific_journal-3517 1 37 and and CCONJ american_scientific_journal-3517 1 38 researchers researcher NOUN american_scientific_journal-3517 1 39 http://asrjetsjournal.org/ http://asrjetsjournal.org/ VERB american_scientific_journal-3517 1 40 the the DET american_scientific_journal-3517 1 41 relationship relationship NOUN american_scientific_journal-3517 1 42 of of ADP american_scientific_journal-3517 1 43 generalized generalized ADJ american_scientific_journal-3517 1 44 fractional fractional ADJ american_scientific_journal-3517 1 45 hilbert hilbert NOUN american_scientific_journal-3517 1 46 transform transform NOUN american_scientific_journal-3517 1 47 with with ADP american_scientific_journal-3517 1 48 fractional fractional ADJ american_scientific_journal-3517 1 49 mellin mellin PROPN american_scientific_journal-3517 1 50 and and CCONJ american_scientific_journal-3517 1 51 fractional fractional ADJ american_scientific_journal-3517 1 52 laplace laplace NOUN american_scientific_journal-3517 1 53 transforms transform VERB american_scientific_journal-3517 1 54 laljahana laljahana ADJ american_scientific_journal-3517 1 55 , , PUNCT american_scientific_journal-3517 1 56 saleem saleem PROPN american_scientific_journal-3517 1 57 iqbalb iqbalb PROPN american_scientific_journal-3517 1 58 * * SYM american_scientific_journal-3517 1 59 , , PUNCT american_scientific_journal-3517 1 60 farhana farhana PROPN american_scientific_journal-3517 1 61 sarwarc sarwarc PROPN american_scientific_journal-3517 1 62 , , PUNCT american_scientific_journal-3517 1 63 syed syed PROPN american_scientific_journal-3517 1 64 mohsin mohsin PROPN american_scientific_journal-3517 1 65 razad razad PROPN american_scientific_journal-3517 1 66 , , PUNCT american_scientific_journal-3517 1 67 abdul abdul PROPN american_scientific_journal-3517 1 68 rehmane rehmane PROPN american_scientific_journal-3517 1 69 a a DET american_scientific_journal-3517 1 70 , , PUNCT american_scientific_journal-3517 1 71 b b NOUN american_scientific_journal-3517 1 72 , , PUNCT american_scientific_journal-3517 1 73 edepartment edepartment NOUN american_scientific_journal-3517 1 74 of of ADP american_scientific_journal-3517 1 75 mathematics mathematic NOUN american_scientific_journal-3517 1 76 , , PUNCT american_scientific_journal-3517 1 77 university university NOUN american_scientific_journal-3517 1 78 of of ADP american_scientific_journal-3517 1 79 balochistan balochistan PROPN american_scientific_journal-3517 1 80 , , PUNCT american_scientific_journal-3517 1 81 quetta quetta PROPN american_scientific_journal-3517 1 82 87300 87300 NUM american_scientific_journal-3517 1 83 , , PUNCT american_scientific_journal-3517 1 84 pakistan pakistan PROPN american_scientific_journal-3517 1 85 cdepartment cdepartment NOUN american_scientific_journal-3517 1 86 of of ADP american_scientific_journal-3517 1 87 mathematics mathematics PROPN american_scientific_journal-3517 1 88 f.g.girls f.g.girl NOUN american_scientific_journal-3517 1 89 degree degree NOUN american_scientific_journal-3517 1 90 college college NOUN american_scientific_journal-3517 1 91 , , PUNCT american_scientific_journal-3517 1 92 madrissa madrissa NOUN american_scientific_journal-3517 1 93 road road NOUN american_scientific_journal-3517 1 94 , , PUNCT american_scientific_journal-3517 1 95 quetta quetta PROPN american_scientific_journal-3517 1 96 , , PUNCT american_scientific_journal-3517 1 97 cantt cantt PROPN american_scientific_journal-3517 1 98 , , PUNCT american_scientific_journal-3517 1 99 87300 87300 NUM american_scientific_journal-3517 1 100 , , PUNCT american_scientific_journal-3517 1 101 pakistan pakistan PROPN american_scientific_journal-3517 1 102 ddepartmen ddepartman NOUN american_scientific_journal-3517 1 103 of of ADP american_scientific_journal-3517 1 104 physics physics PROPN american_scientific_journal-3517 1 105 , , PUNCT american_scientific_journal-3517 1 106 university university NOUN american_scientific_journal-3517 1 107 of of ADP american_scientific_journal-3517 1 108 balochistan balochistan PROPN american_scientific_journal-3517 1 109 , , PUNCT american_scientific_journal-3517 1 110 quetta quetta PROPN american_scientific_journal-3517 1 111 87300 87300 NUM american_scientific_journal-3517 1 112 , , PUNCT american_scientific_journal-3517 1 113 pakistan pakistan PROPN american_scientific_journal-3517 1 114 aemail aemail NOUN american_scientific_journal-3517 1 115 : : PUNCT american_scientific_journal-3517 2 1 laljahanbaloch@gmail.com laljahanbaloch@gmail.com X american_scientific_journal-3517 2 2 bemail bemail NOUN american_scientific_journal-3517 2 3 : : PUNCT american_scientific_journal-3517 2 4 saleemiqbal81@yahoo.com saleemiqbal81@yahoo.com X american_scientific_journal-3517 2 5 cemail cemail NOUN american_scientific_journal-3517 2 6 : : PUNCT american_scientific_journal-3517 2 7 f_saleem10@yahoo.com f_saleem10@yahoo.com X american_scientific_journal-3517 2 8 demail demail NOUN american_scientific_journal-3517 2 9 : : PUNCT american_scientific_journal-3517 2 10 smraza7@yahoo.com smraza7@yahoo.com X american_scientific_journal-3517 2 11 eemail eemail NOUN american_scientific_journal-3517 2 12 : : PUNCT american_scientific_journal-3517 2 13 abdul_maths@yahoo.com abdul_maths@yahoo.com X american_scientific_journal-3517 3 1 abstract abstract ADJ american_scientific_journal-3517 3 2 we we PRON american_scientific_journal-3517 3 3 have have AUX american_scientific_journal-3517 3 4 developed develop VERB american_scientific_journal-3517 3 5 in in ADP american_scientific_journal-3517 3 6 this this DET american_scientific_journal-3517 3 7 research research NOUN american_scientific_journal-3517 3 8 paper paper NOUN american_scientific_journal-3517 3 9 , , PUNCT american_scientific_journal-3517 3 10 some some PRON american_scientific_journal-3517 3 11 of of ADP american_scientific_journal-3517 3 12 the the DET american_scientific_journal-3517 3 13 fundamental fundamental ADJ american_scientific_journal-3517 3 14 relationship relationship NOUN american_scientific_journal-3517 3 15 between between ADP american_scientific_journal-3517 3 16 generalized generalized ADJ american_scientific_journal-3517 3 17 fractional fractional ADJ american_scientific_journal-3517 3 18 hilbert hilbert NOUN american_scientific_journal-3517 3 19 transform transform NOUN american_scientific_journal-3517 3 20 with with ADP american_scientific_journal-3517 3 21 fractional fractional ADJ american_scientific_journal-3517 3 22 mellin mellin PROPN american_scientific_journal-3517 3 23 transform transform NOUN american_scientific_journal-3517 3 24 , , PUNCT american_scientific_journal-3517 3 25 fractional fractional ADJ american_scientific_journal-3517 3 26 laplace laplace NOUN american_scientific_journal-3517 3 27 transform transform NOUN american_scientific_journal-3517 3 28 , , PUNCT american_scientific_journal-3517 3 29 fractional fractional ADJ american_scientific_journal-3517 3 30 inverse inverse ADJ american_scientific_journal-3517 3 31 laplace laplace NOUN american_scientific_journal-3517 3 32 transform transform NOUN american_scientific_journal-3517 3 33 .. .. PUNCT american_scientific_journal-3517 4 1 the the DET american_scientific_journal-3517 4 2 results result NOUN american_scientific_journal-3517 4 3 are be AUX american_scientific_journal-3517 4 4 mathematically mathematically ADV american_scientific_journal-3517 4 5 expressed express VERB american_scientific_journal-3517 4 6 . . PUNCT american_scientific_journal-3517 5 1 these these DET american_scientific_journal-3517 5 2 results result NOUN american_scientific_journal-3517 5 3 , , PUNCT american_scientific_journal-3517 5 4 however however ADV american_scientific_journal-3517 5 5 , , PUNCT american_scientific_journal-3517 5 6 need need VERB american_scientific_journal-3517 5 7 modelling modelling NOUN american_scientific_journal-3517 5 8 and and CCONJ american_scientific_journal-3517 5 9 simulation simulation NOUN american_scientific_journal-3517 5 10 with with ADP american_scientific_journal-3517 5 11 any any DET american_scientific_journal-3517 5 12 specialized specialized ADJ american_scientific_journal-3517 5 13 signal signal NOUN american_scientific_journal-3517 5 14 processing processing NOUN american_scientific_journal-3517 5 15 data datum NOUN american_scientific_journal-3517 5 16 . . PUNCT american_scientific_journal-3517 6 1 keywords keyword NOUN american_scientific_journal-3517 6 2 : : PUNCT american_scientific_journal-3517 6 3 fractional fractional ADJ american_scientific_journal-3517 6 4 hilbert hilbert NOUN american_scientific_journal-3517 6 5 transform transform NOUN american_scientific_journal-3517 6 6 ; ; PUNCT american_scientific_journal-3517 6 7 fractional fractional ADJ american_scientific_journal-3517 6 8 mellin mellin PROPN american_scientific_journal-3517 6 9 transform transform NOUN american_scientific_journal-3517 6 10 ; ; PUNCT american_scientific_journal-3517 6 11 fractional fractional ADJ american_scientific_journal-3517 6 12 laplace laplace NOUN american_scientific_journal-3517 6 13 transform transform NOUN american_scientific_journal-3517 6 14 ; ; PUNCT american_scientific_journal-3517 6 15 fractional fractional ADJ american_scientific_journal-3517 6 16 fourier fourier NOUN american_scientific_journal-3517 6 17 transform transform NOUN american_scientific_journal-3517 6 18 . . PUNCT american_scientific_journal-3517 7 1 1 1 X american_scientific_journal-3517 7 2 . . X american_scientific_journal-3517 7 3 introduction introduction NOUN american_scientific_journal-3517 7 4 fractional fractional ADJ american_scientific_journal-3517 7 5 hilbert hilbert PROPN american_scientific_journal-3517 7 6 transform(frht transform(frht PROPN american_scientific_journal-3517 7 7 ) ) PUNCT american_scientific_journal-3517 7 8 is be AUX american_scientific_journal-3517 7 9 introduced introduce VERB american_scientific_journal-3517 7 10 by by ADP american_scientific_journal-3517 7 11 lohmann lohmann PROPN american_scientific_journal-3517 7 12 and and CCONJ american_scientific_journal-3517 7 13 his his PRON american_scientific_journal-3517 7 14 colleagues colleague NOUN american_scientific_journal-3517 7 15 [ [ X american_scientific_journal-3517 7 16 1 1 NUM american_scientific_journal-3517 7 17 ] ] PUNCT american_scientific_journal-3517 7 18 . . PUNCT american_scientific_journal-3517 8 1 they they PRON american_scientific_journal-3517 8 2 proved prove VERB american_scientific_journal-3517 8 3 in in ADP american_scientific_journal-3517 8 4 their their PRON american_scientific_journal-3517 8 5 paper paper NOUN american_scientific_journal-3517 8 6 that that SCONJ american_scientific_journal-3517 8 7 the the DET american_scientific_journal-3517 8 8 frht frht NOUN american_scientific_journal-3517 8 9 is be AUX american_scientific_journal-3517 8 10 the the DET american_scientific_journal-3517 8 11 generalization generalization NOUN american_scientific_journal-3517 8 12 of of ADP american_scientific_journal-3517 8 13 the the DET american_scientific_journal-3517 8 14 hilbert hilbert NOUN american_scientific_journal-3517 8 15 transform(ht transform(ht NOUN american_scientific_journal-3517 8 16 ) ) PUNCT american_scientific_journal-3517 8 17 . . PUNCT american_scientific_journal-3517 9 1 their their PRON american_scientific_journal-3517 9 2 generalization generalization NOUN american_scientific_journal-3517 9 3 is be AUX american_scientific_journal-3517 9 4 based base VERB american_scientific_journal-3517 9 5 on on ADP american_scientific_journal-3517 9 6 modifying modify VERB american_scientific_journal-3517 9 7 the the DET american_scientific_journal-3517 9 8 spatial spatial ADJ american_scientific_journal-3517 9 9 filters filter NOUN american_scientific_journal-3517 9 10 and and CCONJ american_scientific_journal-3517 9 11 fractional fractional ADJ american_scientific_journal-3517 9 12 fourier fourier NOUN american_scientific_journal-3517 9 13 plane plane NOUN american_scientific_journal-3517 9 14 for for ADP american_scientific_journal-3517 9 15 filters filter NOUN american_scientific_journal-3517 9 16 . . PUNCT american_scientific_journal-3517 10 1 ----------------------------------------------------------------------- ----------------------------------------------------------------------- PUNCT american_scientific_journal-3517 11 1 * * PUNCT american_scientific_journal-3517 11 2 corresponding correspond VERB american_scientific_journal-3517 11 3 author author NOUN american_scientific_journal-3517 11 4 . . PUNCT american_scientific_journal-3517 12 1 http://asrjetsjournal.org/ http://asrjetsjournal.org/ PROPN american_scientific_journal-3517 12 2 american american PROPN american_scientific_journal-3517 12 3 scientific scientific PROPN american_scientific_journal-3517 12 4 research research NOUN american_scientific_journal-3517 12 5 journal journal NOUN american_scientific_journal-3517 12 6 for for ADP american_scientific_journal-3517 12 7 engineering engineering NOUN american_scientific_journal-3517 12 8 , , PUNCT american_scientific_journal-3517 12 9 technology technology NOUN american_scientific_journal-3517 12 10 , , PUNCT american_scientific_journal-3517 12 11 and and CCONJ american_scientific_journal-3517 12 12 sciences science NOUN american_scientific_journal-3517 12 13 ( ( PUNCT american_scientific_journal-3517 12 14 asrjets asrjet NOUN american_scientific_journal-3517 12 15 ) ) PUNCT american_scientific_journal-3517 12 16 ( ( PUNCT american_scientific_journal-3517 12 17 2017 2017 NUM american_scientific_journal-3517 12 18 ) ) PUNCT american_scientific_journal-3517 12 19 volume volume NOUN american_scientific_journal-3517 12 20 38 38 NUM american_scientific_journal-3517 12 21 , , PUNCT american_scientific_journal-3517 12 22 no no DET american_scientific_journal-3517 12 23 2 2 NUM american_scientific_journal-3517 12 24 , , PUNCT american_scientific_journal-3517 12 25 pp pp ADV american_scientific_journal-3517 12 26 90 90 NUM american_scientific_journal-3517 12 27 - - SYM american_scientific_journal-3517 12 28 97 97 NUM american_scientific_journal-3517 12 29 91 91 NUM american_scientific_journal-3517 12 30 in in ADP american_scientific_journal-3517 12 31 continuation continuation NOUN american_scientific_journal-3517 12 32 of of ADP american_scientific_journal-3517 12 33 the the DET american_scientific_journal-3517 12 34 work work NOUN american_scientific_journal-3517 12 35 [ [ X american_scientific_journal-3517 12 36 1 1 NUM american_scientific_journal-3517 12 37 ] ] PUNCT american_scientific_journal-3517 12 38 , , PUNCT american_scientific_journal-3517 12 39 davis davis PROPN american_scientific_journal-3517 12 40 and and CCONJ american_scientific_journal-3517 12 41 his his PRON american_scientific_journal-3517 12 42 colleagues colleague NOUN american_scientific_journal-3517 13 1 [ [ X american_scientific_journal-3517 13 2 2 2 NUM american_scientific_journal-3517 13 3 ] ] PUNCT american_scientific_journal-3517 13 4 employed employ VERB american_scientific_journal-3517 13 5 frft frft ADJ american_scientific_journal-3517 13 6 for for ADP american_scientific_journal-3517 13 7 edge edge NOUN american_scientific_journal-3517 13 8 detection detection NOUN american_scientific_journal-3517 13 9 and and CCONJ american_scientific_journal-3517 13 10 proved prove VERB american_scientific_journal-3517 13 11 that that SCONJ american_scientific_journal-3517 13 12 as as SCONJ american_scientific_journal-3517 13 13 the the DET american_scientific_journal-3517 13 14 fractional fractional ADJ american_scientific_journal-3517 13 15 order order NOUN american_scientific_journal-3517 13 16 is be AUX american_scientific_journal-3517 13 17 varied varied ADJ american_scientific_journal-3517 13 18 then then ADV american_scientific_journal-3517 13 19 different different ADJ american_scientific_journal-3517 13 20 qualities quality NOUN american_scientific_journal-3517 13 21 of of ADP american_scientific_journal-3517 13 22 edge edge NOUN american_scientific_journal-3517 13 23 enhancement enhancement NOUN american_scientific_journal-3517 13 24 are be AUX american_scientific_journal-3517 13 25 obtained obtain VERB american_scientific_journal-3517 13 26 . . PUNCT american_scientific_journal-3517 14 1 in in ADP american_scientific_journal-3517 14 2 [ [ X american_scientific_journal-3517 14 3 3 3 NUM american_scientific_journal-3517 14 4 ] ] SYM american_scientific_journal-3517 14 5 discrete discrete ADJ american_scientific_journal-3517 14 6 counterpart counterpart NOUN american_scientific_journal-3517 14 7 of of ADP american_scientific_journal-3517 14 8 the the DET american_scientific_journal-3517 14 9 frht frht NOUN american_scientific_journal-3517 14 10 is be AUX american_scientific_journal-3517 14 11 discussed discuss VERB american_scientific_journal-3517 14 12 . . PUNCT american_scientific_journal-3517 15 1 zayed zaye VERB american_scientific_journal-3517 16 1 [ [ X american_scientific_journal-3517 16 2 4 4 X american_scientific_journal-3517 16 3 ] ] PUNCT american_scientific_journal-3517 16 4 took take VERB american_scientific_journal-3517 16 5 the the DET american_scientific_journal-3517 16 6 analytical analytical ADJ american_scientific_journal-3517 16 7 signal signal NOUN american_scientific_journal-3517 16 8 ( ( PUNCT american_scientific_journal-3517 16 9 as as ADP american_scientific_journal-3517 16 10 ) ) PUNCT american_scientific_journal-3517 16 11 formalism formalism NOUN american_scientific_journal-3517 16 12 associated associate VERB american_scientific_journal-3517 16 13 with with ADP american_scientific_journal-3517 16 14 the the DET american_scientific_journal-3517 16 15 standard standard ADJ american_scientific_journal-3517 16 16 fourier fourier NOUN american_scientific_journal-3517 16 17 transform transform NOUN american_scientific_journal-3517 16 18 ( ( PUNCT american_scientific_journal-3517 16 19 ft ft NOUN american_scientific_journal-3517 16 20 ) ) PUNCT american_scientific_journal-3517 16 21 and and CCONJ american_scientific_journal-3517 16 22 provided provide VERB american_scientific_journal-3517 16 23 a a DET american_scientific_journal-3517 16 24 counterpart counterpart NOUN american_scientific_journal-3517 16 25 of of ADP american_scientific_journal-3517 16 26 it it PRON american_scientific_journal-3517 16 27 for for ADP american_scientific_journal-3517 16 28 the the DET american_scientific_journal-3517 16 29 fractional fractional ADJ american_scientific_journal-3517 16 30 fourier fourier NOUN american_scientific_journal-3517 16 31 transform transform NOUN american_scientific_journal-3517 16 32 ( ( PUNCT american_scientific_journal-3517 16 33 frft frft ADJ american_scientific_journal-3517 16 34 ) ) PUNCT american_scientific_journal-3517 16 35 . . PUNCT american_scientific_journal-3517 17 1 the the DET american_scientific_journal-3517 17 2 communication communication NOUN american_scientific_journal-3517 17 3 applications application NOUN american_scientific_journal-3517 17 4 is be AUX american_scientific_journal-3517 17 5 discussed discuss VERB american_scientific_journal-3517 17 6 by by ADP american_scientific_journal-3517 17 7 using use VERB american_scientific_journal-3517 17 8 frht frht NOUN american_scientific_journal-3517 17 9 in in ADP american_scientific_journal-3517 17 10 [ [ X american_scientific_journal-3517 17 11 5 5 NUM american_scientific_journal-3517 17 12 ] ] X american_scientific_journal-3517 17 13 fractional fractional ADJ american_scientific_journal-3517 17 14 hilbert hilbert NOUN american_scientific_journal-3517 17 15 transform transform NOUN american_scientific_journal-3517 17 16 is be AUX american_scientific_journal-3517 17 17 a a DET american_scientific_journal-3517 17 18 special special ADJ american_scientific_journal-3517 17 19 case case NOUN american_scientific_journal-3517 17 20 of of ADP american_scientific_journal-3517 17 21 linear linear ADJ american_scientific_journal-3517 17 22 canonical canonical ADJ american_scientific_journal-3517 17 23 transform transform NOUN american_scientific_journal-3517 17 24 ( ( PUNCT american_scientific_journal-3517 17 25 lct lct PROPN american_scientific_journal-3517 17 26 ) ) PUNCT american_scientific_journal-3517 17 27 which which PRON american_scientific_journal-3517 17 28 has have VERB american_scientific_journal-3517 17 29 diverse diverse ADJ american_scientific_journal-3517 17 30 applications application NOUN american_scientific_journal-3517 17 31 such such ADJ american_scientific_journal-3517 17 32 as as ADP american_scientific_journal-3517 17 33 in in ADP american_scientific_journal-3517 17 34 an an DET american_scientific_journal-3517 17 35 image image NOUN american_scientific_journal-3517 17 36 enhancement enhancement NOUN american_scientific_journal-3517 17 37 or or CCONJ american_scientific_journal-3517 17 38 compression compression NOUN american_scientific_journal-3517 17 39 by by ADP american_scientific_journal-3517 17 40 using use VERB american_scientific_journal-3517 17 41 the the DET american_scientific_journal-3517 17 42 angle angle NOUN american_scientific_journal-3517 17 43 of of ADP american_scientific_journal-3517 17 44 rotation rotation NOUN american_scientific_journal-3517 17 45 in in ADP american_scientific_journal-3517 17 46 the the DET american_scientific_journal-3517 17 47 complex complex ADJ american_scientific_journal-3517 17 48 plane plane NOUN american_scientific_journal-3517 17 49 ( ( PUNCT american_scientific_journal-3517 17 50 t t PROPN american_scientific_journal-3517 17 51 , , PUNCT american_scientific_journal-3517 17 52 w w NOUN american_scientific_journal-3517 17 53 ) ) PUNCT american_scientific_journal-3517 17 54 of of ADP american_scientific_journal-3517 17 55 fractional fractional ADJ american_scientific_journal-3517 17 56 fourier fourier NOUN american_scientific_journal-3517 17 57 transform transform NOUN american_scientific_journal-3517 17 58 ( ( PUNCT american_scientific_journal-3517 17 59 frft frft ADJ american_scientific_journal-3517 17 60 ) ) PUNCT american_scientific_journal-3517 17 61 on on ADP american_scientific_journal-3517 17 62 optical optical ADJ american_scientific_journal-3517 17 63 systems system NOUN american_scientific_journal-3517 17 64 , , PUNCT american_scientific_journal-3517 17 65 edge edge NOUN american_scientific_journal-3517 17 66 deduction deduction NOUN american_scientific_journal-3517 17 67 for for ADP american_scientific_journal-3517 17 68 propagation propagation NOUN american_scientific_journal-3517 17 69 and and CCONJ american_scientific_journal-3517 17 70 delay delay NOUN american_scientific_journal-3517 17 71 times time NOUN american_scientific_journal-3517 17 72 , , PUNCT american_scientific_journal-3517 17 73 beam beam NOUN american_scientific_journal-3517 17 74 flowing flow VERB american_scientific_journal-3517 17 75 and and CCONJ american_scientific_journal-3517 17 76 indeed indeed ADV american_scientific_journal-3517 17 77 in in ADP american_scientific_journal-3517 17 78 signal signal ADJ american_scientific_journal-3517 17 79 processing processing NOUN american_scientific_journal-3517 17 80 [ [ X american_scientific_journal-3517 17 81 6 6 NUM american_scientific_journal-3517 17 82 - - SYM american_scientific_journal-3517 17 83 13 13 NUM american_scientific_journal-3517 17 84 ] ] PUNCT american_scientific_journal-3517 17 85 . . PUNCT american_scientific_journal-3517 18 1 needless needless ADJ american_scientific_journal-3517 18 2 to to PART american_scientific_journal-3517 18 3 mention mention VERB american_scientific_journal-3517 18 4 , , PUNCT american_scientific_journal-3517 18 5 the the DET american_scientific_journal-3517 18 6 integral integral ADJ american_scientific_journal-3517 18 7 transforms transform NOUN american_scientific_journal-3517 18 8 have have VERB american_scientific_journal-3517 18 9 significant significant ADJ american_scientific_journal-3517 18 10 applications application NOUN american_scientific_journal-3517 18 11 in in ADP american_scientific_journal-3517 18 12 both both DET american_scientific_journal-3517 18 13 physics physics NOUN american_scientific_journal-3517 18 14 and and CCONJ american_scientific_journal-3517 18 15 applied apply VERB american_scientific_journal-3517 18 16 mathematics mathematic NOUN american_scientific_journal-3517 18 17 . . PUNCT american_scientific_journal-3517 19 1 akilahmad akilahmad ADJ american_scientific_journal-3517 19 2 sheikh sheikh NOUN american_scientific_journal-3517 19 3 and and CCONJ american_scientific_journal-3517 19 4 alka alka PROPN american_scientific_journal-3517 19 5 gudadhe gudadhe PROPN american_scientific_journal-3517 20 1 [ [ X american_scientific_journal-3517 20 2 14,15 14,15 NUM american_scientific_journal-3517 20 3 ] ] PUNCT american_scientific_journal-3517 20 4 worked work VERB american_scientific_journal-3517 20 5 on on ADP american_scientific_journal-3517 20 6 relationships relationship NOUN american_scientific_journal-3517 20 7 on on ADP american_scientific_journal-3517 20 8 generalized generalized ADJ american_scientific_journal-3517 20 9 fractional fractional ADJ american_scientific_journal-3517 20 10 hilbert hilbert NOUN american_scientific_journal-3517 20 11 transform transform NOUN american_scientific_journal-3517 20 12 with with ADP american_scientific_journal-3517 20 13 some some DET american_scientific_journal-3517 20 14 classical classical ADJ american_scientific_journal-3517 20 15 transforms transform NOUN american_scientific_journal-3517 20 16 and and CCONJ american_scientific_journal-3517 20 17 developed develop VERB american_scientific_journal-3517 20 18 analytic analytic ADJ american_scientific_journal-3517 20 19 theorems theorem NOUN american_scientific_journal-3517 20 20 . . PUNCT american_scientific_journal-3517 21 1 the the DET american_scientific_journal-3517 21 2 fractional fractional ADJ american_scientific_journal-3517 21 3 hilbert hilbert NOUN american_scientific_journal-3517 21 4 transform transform NOUN american_scientific_journal-3517 21 5 is be AUX american_scientific_journal-3517 21 6 a a DET american_scientific_journal-3517 21 7 generalization generalization NOUN american_scientific_journal-3517 21 8 of of ADP american_scientific_journal-3517 21 9 hilbert hilbert PROPN american_scientific_journal-3517 21 10 transform transform NOUN american_scientific_journal-3517 21 11 and and CCONJ american_scientific_journal-3517 21 12 so so ADV american_scientific_journal-3517 21 13 is be AUX american_scientific_journal-3517 21 14 the the DET american_scientific_journal-3517 21 15 case case NOUN american_scientific_journal-3517 21 16 of of ADP american_scientific_journal-3517 21 17 fractional fractional ADJ american_scientific_journal-3517 21 18 fourier fourier NOUN american_scientific_journal-3517 21 19 tansform tansform NOUN american_scientific_journal-3517 21 20 ( ( PUNCT american_scientific_journal-3517 21 21 frft frft ADJ american_scientific_journal-3517 21 22 ) ) PUNCT american_scientific_journal-3517 21 23 . . PUNCT american_scientific_journal-3517 22 1 the the DET american_scientific_journal-3517 22 2 scale scale NOUN american_scientific_journal-3517 22 3 invariance invariance NOUN american_scientific_journal-3517 22 4 property property NOUN american_scientific_journal-3517 22 5 of of ADP american_scientific_journal-3517 22 6 fractional fractional ADJ american_scientific_journal-3517 22 7 mellin mellin PROPN american_scientific_journal-3517 22 8 transform transform NOUN american_scientific_journal-3517 22 9 ( ( PUNCT american_scientific_journal-3517 22 10 fmt fmt NOUN american_scientific_journal-3517 22 11 ) ) PUNCT american_scientific_journal-3517 22 12 is be AUX american_scientific_journal-3517 22 13 a a DET american_scientific_journal-3517 22 14 very very ADV american_scientific_journal-3517 22 15 important important ADJ american_scientific_journal-3517 22 16 tool tool NOUN american_scientific_journal-3517 22 17 to to ADP american_scientific_journal-3517 22 18 constructing construct VERB american_scientific_journal-3517 22 19 two two NUM american_scientific_journal-3517 22 20 dimensional dimensional ADJ american_scientific_journal-3517 22 21 real real ADJ american_scientific_journal-3517 22 22 images image NOUN american_scientific_journal-3517 22 23 [ [ X american_scientific_journal-3517 22 24 16 16 NUM american_scientific_journal-3517 22 25 ] ] PUNCT american_scientific_journal-3517 22 26 . . PUNCT american_scientific_journal-3517 23 1 analysis analysis NOUN american_scientific_journal-3517 23 2 of of ADP american_scientific_journal-3517 23 3 hilbert hilbert PROPN american_scientific_journal-3517 23 4 transforms transform VERB american_scientific_journal-3517 23 5 with with ADP american_scientific_journal-3517 23 6 fractional fractional ADJ american_scientific_journal-3517 23 7 fourier fourier NOUN american_scientific_journal-3517 23 8 transform transform NOUN american_scientific_journal-3517 23 9 ( ( PUNCT american_scientific_journal-3517 23 10 frft frft ADJ american_scientific_journal-3517 23 11 ) ) PUNCT american_scientific_journal-3517 24 1 [ [ X american_scientific_journal-3517 24 2 17 17 NUM american_scientific_journal-3517 24 3 - - SYM american_scientific_journal-3517 24 4 18 18 NUM american_scientific_journal-3517 24 5 ] ] PUNCT american_scientific_journal-3517 24 6 resulted result VERB american_scientific_journal-3517 24 7 into into ADP american_scientific_journal-3517 24 8 many many ADJ american_scientific_journal-3517 24 9 fascinating fascinating ADJ american_scientific_journal-3517 24 10 results result NOUN american_scientific_journal-3517 24 11 . . PUNCT american_scientific_journal-3517 25 1 the the DET american_scientific_journal-3517 25 2 use use NOUN american_scientific_journal-3517 25 3 of of ADP american_scientific_journal-3517 25 4 fractional fractional ADJ american_scientific_journal-3517 25 5 hilbert hilbert NOUN american_scientific_journal-3517 25 6 transforms transform VERB american_scientific_journal-3517 25 7 and and CCONJ american_scientific_journal-3517 25 8 its its PRON american_scientific_journal-3517 25 9 extensions extension NOUN american_scientific_journal-3517 25 10 lead lead VERB american_scientific_journal-3517 25 11 to to ADP american_scientific_journal-3517 25 12 analytical analytical ADJ american_scientific_journal-3517 25 13 behavior behavior NOUN american_scientific_journal-3517 25 14 of of ADP american_scientific_journal-3517 25 15 signal signal ADJ american_scientific_journal-3517 25 16 constructions construction NOUN american_scientific_journal-3517 25 17 . . PUNCT american_scientific_journal-3517 26 1 [ [ X american_scientific_journal-3517 26 2 9 9 NUM american_scientific_journal-3517 26 3 ] ] PUNCT american_scientific_journal-3517 26 4 . . PUNCT american_scientific_journal-3517 27 1 this this PRON american_scientific_journal-3517 27 2 is be AUX american_scientific_journal-3517 27 3 how how SCONJ american_scientific_journal-3517 27 4 amplitude amplitude NOUN american_scientific_journal-3517 27 5 modulation modulation NOUN american_scientific_journal-3517 27 6 ( ( PUNCT american_scientific_journal-3517 27 7 am am NOUN american_scientific_journal-3517 27 8 , , PUNCT american_scientific_journal-3517 27 9 ) ) PUNCT american_scientific_journal-3517 27 10 frequency frequency NOUN american_scientific_journal-3517 27 11 modulation modulation NOUN american_scientific_journal-3517 27 12 ( ( PUNCT american_scientific_journal-3517 27 13 fm fm NOUN american_scientific_journal-3517 27 14 ) ) PUNCT american_scientific_journal-3517 27 15 or or CCONJ american_scientific_journal-3517 27 16 phase phase NOUN american_scientific_journal-3517 27 17 modulation(pm modulation(pm NOUN american_scientific_journal-3517 27 18 ) ) PUNCT american_scientific_journal-3517 27 19 can can AUX american_scientific_journal-3517 27 20 be be AUX american_scientific_journal-3517 27 21 exploited exploit VERB american_scientific_journal-3517 27 22 in in ADP american_scientific_journal-3517 27 23 weiner weiner NOUN american_scientific_journal-3517 27 24 filter filter NOUN american_scientific_journal-3517 27 25 for for ADP american_scientific_journal-3517 27 26 controlling control VERB american_scientific_journal-3517 27 27 the the DET american_scientific_journal-3517 27 28 changes change NOUN american_scientific_journal-3517 27 29 in in ADP american_scientific_journal-3517 27 30 phase phase NOUN american_scientific_journal-3517 27 31 signals signal NOUN american_scientific_journal-3517 27 32 . . PUNCT american_scientific_journal-3517 28 1 these these DET american_scientific_journal-3517 28 2 fractional fractional ADJ american_scientific_journal-3517 28 3 phase phase NOUN american_scientific_journal-3517 28 4 changes change NOUN american_scientific_journal-3517 28 5 are be AUX american_scientific_journal-3517 28 6 the the DET american_scientific_journal-3517 28 7 manifestations manifestation NOUN american_scientific_journal-3517 28 8 of of ADP american_scientific_journal-3517 28 9 the the DET american_scientific_journal-3517 28 10 fractional fractional ADJ american_scientific_journal-3517 28 11 hilbert hilbert NOUN american_scientific_journal-3517 28 12 transform transform NOUN american_scientific_journal-3517 28 13 and and CCONJ american_scientific_journal-3517 28 14 of of ADP american_scientific_journal-3517 28 15 the the DET american_scientific_journal-3517 28 16 morlet morlet NOUN american_scientific_journal-3517 28 17 and and CCONJ american_scientific_journal-3517 28 18 harlet harlet NOUN american_scientific_journal-3517 28 19 wavelets wavelet NOUN american_scientific_journal-3517 28 20 for for ADP american_scientific_journal-3517 28 21 two two NUM american_scientific_journal-3517 28 22 super super ADV american_scientific_journal-3517 28 23 posed pose VERB american_scientific_journal-3517 28 24 wave wave NOUN american_scientific_journal-3517 28 25 form form NOUN american_scientific_journal-3517 28 26 . . PUNCT american_scientific_journal-3517 29 1 the the DET american_scientific_journal-3517 29 2 same same ADJ american_scientific_journal-3517 29 3 behavior behavior NOUN american_scientific_journal-3517 29 4 is be AUX american_scientific_journal-3517 29 5 witnessed witness VERB american_scientific_journal-3517 29 6 with with ADP american_scientific_journal-3517 29 7 fourier fouri ADJ american_scientific_journal-3517 29 8 transform transform NOUN american_scientific_journal-3517 29 9 [ [ X american_scientific_journal-3517 29 10 4 4 NUM american_scientific_journal-3517 29 11 ] ] PUNCT american_scientific_journal-3517 29 12 . . PUNCT american_scientific_journal-3517 30 1 the the DET american_scientific_journal-3517 30 2 fractional fractional ADJ american_scientific_journal-3517 30 3 laplace laplace NOUN american_scientific_journal-3517 30 4 transform transform NOUN american_scientific_journal-3517 30 5 is be AUX american_scientific_journal-3517 30 6 also also ADV american_scientific_journal-3517 30 7 the the DET american_scientific_journal-3517 30 8 genialized genialized ADJ american_scientific_journal-3517 30 9 case case NOUN american_scientific_journal-3517 30 10 of of ADP american_scientific_journal-3517 30 11 laplace laplace NOUN american_scientific_journal-3517 30 12 transform transform NOUN american_scientific_journal-3517 30 13 . . PUNCT american_scientific_journal-3517 31 1 heaviside heaviside ADJ american_scientific_journal-3517 31 2 step step NOUN american_scientific_journal-3517 31 3 function function NOUN american_scientific_journal-3517 31 4 is be AUX american_scientific_journal-3517 31 5 a a DET american_scientific_journal-3517 31 6 step step NOUN american_scientific_journal-3517 31 7 forward forward ADV american_scientific_journal-3517 31 8 to to PART american_scientific_journal-3517 31 9 deal deal VERB american_scientific_journal-3517 31 10 with with ADP american_scientific_journal-3517 31 11 laplace laplace NOUN american_scientific_journal-3517 31 12 transform transform NOUN american_scientific_journal-3517 31 13 because because SCONJ american_scientific_journal-3517 31 14 it it PRON american_scientific_journal-3517 31 15 deals deal VERB american_scientific_journal-3517 31 16 in in ADP american_scientific_journal-3517 31 17 the the DET american_scientific_journal-3517 31 18 denominator denominator NOUN american_scientific_journal-3517 31 19 with with ADP american_scientific_journal-3517 31 20 a a DET american_scientific_journal-3517 31 21 function function NOUN american_scientific_journal-3517 31 22 with with ADP american_scientific_journal-3517 31 23 fractional fractional ADJ american_scientific_journal-3517 31 24 exponent exponent NOUN american_scientific_journal-3517 31 25 . . PUNCT american_scientific_journal-3517 32 1 laplace laplace PROPN american_scientific_journal-3517 32 2 and and CCONJ american_scientific_journal-3517 32 3 its its PRON american_scientific_journal-3517 32 4 corresponding corresponding ADJ american_scientific_journal-3517 32 5 fractional fractional ADJ american_scientific_journal-3517 32 6 laplace laplace NOUN american_scientific_journal-3517 32 7 transform transform NOUN american_scientific_journal-3517 32 8 deals deal NOUN american_scientific_journal-3517 32 9 with with ADP american_scientific_journal-3517 32 10 wigner wigner ADJ american_scientific_journal-3517 32 11 distribution(wd distribution(wd NOUN american_scientific_journal-3517 32 12 ) ) PUNCT american_scientific_journal-3517 32 13 , , PUNCT american_scientific_journal-3517 32 14 wiener wiener NOUN american_scientific_journal-3517 32 15 space(ws space(ws PROPN american_scientific_journal-3517 32 16 ) ) PUNCT american_scientific_journal-3517 32 17 the the DET american_scientific_journal-3517 32 18 ambiguity ambiguity NOUN american_scientific_journal-3517 32 19 fraction fraction NOUN american_scientific_journal-3517 32 20 , , PUNCT american_scientific_journal-3517 32 21 the the DET american_scientific_journal-3517 32 22 short short ADJ american_scientific_journal-3517 32 23 time time NOUN american_scientific_journal-3517 32 24 fourier fourier ADV american_scientific_journal-3517 32 25 transform(sstft transform(sstft PROPN american_scientific_journal-3517 32 26 ) ) PUNCT american_scientific_journal-3517 32 27 , , PUNCT american_scientific_journal-3517 32 28 speech speech NOUN american_scientific_journal-3517 32 29 processing processing NOUN american_scientific_journal-3517 32 30 , , PUNCT american_scientific_journal-3517 32 31 radar radar NOUN american_scientific_journal-3517 32 32 , , PUNCT american_scientific_journal-3517 32 33 image image NOUN american_scientific_journal-3517 32 34 rotation rotation NOUN american_scientific_journal-3517 32 35 , , PUNCT american_scientific_journal-3517 32 36 confocal confocal ADJ american_scientific_journal-3517 32 37 microscopy microscopy NOUN american_scientific_journal-3517 32 38 , , PUNCT american_scientific_journal-3517 32 39 etc etc X american_scientific_journal-3517 32 40 . . X american_scientific_journal-3517 33 1 the the DET american_scientific_journal-3517 33 2 aperiodic aperiodic ADJ american_scientific_journal-3517 33 3 stable stable ADJ american_scientific_journal-3517 33 4 chaos chaos NOUN american_scientific_journal-3517 33 5 with with ADP american_scientific_journal-3517 33 6 lyapunov lyapunov ADJ american_scientific_journal-3517 33 7 experiments experiment NOUN american_scientific_journal-3517 33 8 in in ADP american_scientific_journal-3517 33 9 real real ADJ american_scientific_journal-3517 33 10 time time NOUN american_scientific_journal-3517 33 11 signals signal NOUN american_scientific_journal-3517 33 12 can can AUX american_scientific_journal-3517 33 13 be be AUX american_scientific_journal-3517 33 14 studied study VERB american_scientific_journal-3517 33 15 with with ADP american_scientific_journal-3517 33 16 fractional fractional ADJ american_scientific_journal-3517 33 17 laplace laplace NOUN american_scientific_journal-3517 33 18 transform transform NOUN american_scientific_journal-3517 33 19 [ [ X american_scientific_journal-3517 33 20 19 19 NUM american_scientific_journal-3517 33 21 - - SYM american_scientific_journal-3517 33 22 20 20 NUM american_scientific_journal-3517 33 23 ] ] PUNCT american_scientific_journal-3517 33 24 . . PUNCT american_scientific_journal-3517 34 1 2 2 X american_scientific_journal-3517 34 2 . . X american_scientific_journal-3517 34 3 results result NOUN american_scientific_journal-3517 34 4 and and CCONJ american_scientific_journal-3517 34 5 discussions discussion NOUN american_scientific_journal-3517 34 6 2.1 2.1 NUM american_scientific_journal-3517 34 7 . . PUNCT american_scientific_journal-3517 35 1 the the DET american_scientific_journal-3517 35 2 relationship relationship NOUN american_scientific_journal-3517 35 3 between between ADP american_scientific_journal-3517 35 4 generalized generalized ADJ american_scientific_journal-3517 35 5 fractional fractional ADJ american_scientific_journal-3517 35 6 hilbert hilbert NOUN american_scientific_journal-3517 35 7 transform transform NOUN american_scientific_journal-3517 35 8 with with ADP american_scientific_journal-3517 35 9 fractional fractional ADJ american_scientific_journal-3517 35 10 mellin mellin PROPN american_scientific_journal-3517 35 11 transform transform VERB american_scientific_journal-3517 35 12 the the DET american_scientific_journal-3517 35 13 relation relation NOUN american_scientific_journal-3517 35 14 between between ADP american_scientific_journal-3517 35 15 fractional fractional ADJ american_scientific_journal-3517 35 16 mellin mellin PROPN american_scientific_journal-3517 35 17 transform transform NOUN american_scientific_journal-3517 35 18 and and CCONJ american_scientific_journal-3517 35 19 fractional fractional ADJ american_scientific_journal-3517 35 20 fourier fourier NOUN american_scientific_journal-3517 35 21 is be AUX american_scientific_journal-3517 35 22 defined define VERB american_scientific_journal-3517 35 23 in in ADP american_scientific_journal-3517 35 24 [ [ X american_scientific_journal-3517 35 25 16 16 NUM american_scientific_journal-3517 35 26 ] ] PUNCT american_scientific_journal-3517 35 27 as as ADP american_scientific_journal-3517 35 28 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(π‘₯π‘₯ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-3517 35 29 ) ) PUNCT american_scientific_journal-3517 35 30 ] ] PUNCT american_scientific_journal-3517 36 1 = = PUNCT american_scientific_journal-3517 36 2 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(𝑒𝑒𝑑𝑑 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(𝑒𝑒𝑑𝑑 X american_scientific_journal-3517 36 3 ) ) PUNCT american_scientific_journal-3517 36 4 ] ] PUNCT american_scientific_journal-3517 37 1 = = PUNCT american_scientific_journal-3517 37 2 οΏ½ οΏ½ PROPN american_scientific_journal-3517 37 3 1 1 NUM american_scientific_journal-3517 37 4 βˆ’ βˆ’ PROPN american_scientific_journal-3517 37 5 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 37 6 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 37 7 οΏ½ οΏ½ PROPN american_scientific_journal-3517 38 1 𝑓𝑓(𝑒𝑒𝑑𝑑 𝑓𝑓(𝑒𝑒𝑑𝑑 PRON american_scientific_journal-3517 38 2 ) ) PUNCT american_scientific_journal-3517 39 1 ∞ ∞ PROPN american_scientific_journal-3517 39 2 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 39 3 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 39 4 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 39 5 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 39 6 ( ( PUNCT american_scientific_journal-3517 39 7 1 1 X american_scientific_journal-3517 39 8 ) ) PUNCT american_scientific_journal-3517 39 9 frft frft ADJ american_scientific_journal-3517 39 10 is be AUX american_scientific_journal-3517 39 11 the the DET american_scientific_journal-3517 39 12 fractional fractional ADJ american_scientific_journal-3517 39 13 fourier fourier NOUN american_scientific_journal-3517 39 14 transform transform NOUN american_scientific_journal-3517 39 15 and and CCONJ american_scientific_journal-3517 39 16 its its PRON american_scientific_journal-3517 39 17 definition definition NOUN american_scientific_journal-3517 39 18 and and CCONJ american_scientific_journal-3517 39 19 its its PRON american_scientific_journal-3517 39 20 applications application NOUN american_scientific_journal-3517 39 21 are be AUX american_scientific_journal-3517 39 22 disused disuse VERB american_scientific_journal-3517 39 23 in in ADP american_scientific_journal-3517 39 24 [ [ PUNCT american_scientific_journal-3517 39 25 21 21 NUM american_scientific_journal-3517 39 26 - - SYM american_scientific_journal-3517 39 27 23 23 NUM american_scientific_journal-3517 39 28 ] ] PUNCT american_scientific_journal-3517 39 29 . . PUNCT american_scientific_journal-3517 40 1 where where SCONJ american_scientific_journal-3517 40 2 frmt frmt NOUN american_scientific_journal-3517 40 3 is be AUX american_scientific_journal-3517 40 4 fractional fractional ADJ american_scientific_journal-3517 40 5 mellin mellin NOUN american_scientific_journal-3517 40 6 transform transform NOUN american_scientific_journal-3517 40 7 and and CCONJ american_scientific_journal-3517 40 8 defined define VERB american_scientific_journal-3517 40 9 as as ADP american_scientific_journal-3517 40 10 american american PROPN american_scientific_journal-3517 40 11 scientific scientific PROPN american_scientific_journal-3517 40 12 research research PROPN american_scientific_journal-3517 40 13 journal journal NOUN american_scientific_journal-3517 40 14 for for ADP american_scientific_journal-3517 40 15 engineering engineering NOUN american_scientific_journal-3517 40 16 , , PUNCT american_scientific_journal-3517 40 17 technology technology NOUN american_scientific_journal-3517 40 18 , , PUNCT american_scientific_journal-3517 40 19 and and CCONJ american_scientific_journal-3517 40 20 sciences science NOUN american_scientific_journal-3517 40 21 ( ( PUNCT american_scientific_journal-3517 40 22 asrjets asrjet NOUN american_scientific_journal-3517 40 23 ) ) PUNCT american_scientific_journal-3517 40 24 ( ( PUNCT american_scientific_journal-3517 40 25 2017 2017 NUM american_scientific_journal-3517 40 26 ) ) PUNCT american_scientific_journal-3517 40 27 volume volume NOUN american_scientific_journal-3517 40 28 38 38 NUM american_scientific_journal-3517 40 29 , , PUNCT american_scientific_journal-3517 40 30 no no DET american_scientific_journal-3517 40 31 2 2 NUM american_scientific_journal-3517 40 32 , , PUNCT american_scientific_journal-3517 40 33 pp pp ADV american_scientific_journal-3517 40 34 90 90 NUM american_scientific_journal-3517 40 35 - - SYM american_scientific_journal-3517 40 36 97 97 NUM american_scientific_journal-3517 40 37 92 92 NUM american_scientific_journal-3517 40 38 𝐹𝐹𝑖𝑖(𝑒𝑒 𝐹𝐹𝑖𝑖(𝑒𝑒 NUM american_scientific_journal-3517 40 39 ) ) PUNCT american_scientific_journal-3517 40 40 = = SYM american_scientific_journal-3517 40 41 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(𝑖𝑖 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(𝑖𝑖 NUM american_scientific_journal-3517 40 42 ) ) PUNCT american_scientific_journal-3517 40 43 ] ] PUNCT american_scientific_journal-3517 41 1 = = PUNCT american_scientific_journal-3517 41 2 οΏ½ οΏ½ PROPN american_scientific_journal-3517 41 3 1 1 NUM american_scientific_journal-3517 41 4 βˆ’ βˆ’ PROPN american_scientific_journal-3517 41 5 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 41 6 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 41 7 οΏ½ οΏ½ PROPN american_scientific_journal-3517 41 8 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 41 9 ) ) PUNCT american_scientific_journal-3517 41 10 ∞ ∞ PROPN american_scientific_journal-3517 42 1 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 42 2 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 42 3 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 42 4 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 42 5 the the DET american_scientific_journal-3517 42 6 generalize generalize VERB american_scientific_journal-3517 42 7 fractional fractional ADJ american_scientific_journal-3517 42 8 hilbert hilbert NOUN american_scientific_journal-3517 42 9 transform transform NOUN american_scientific_journal-3517 42 10 is be AUX american_scientific_journal-3517 42 11 defined define VERB american_scientific_journal-3517 42 12 in in ADP american_scientific_journal-3517 42 13 [ [ X american_scientific_journal-3517 42 14 14 14 NUM american_scientific_journal-3517 42 15 ] ] PUNCT american_scientific_journal-3517 42 16 as as ADP american_scientific_journal-3517 42 17 𝐻𝐻𝑖𝑖[𝑓𝑓(π‘₯π‘₯)](𝑖𝑖 𝐻𝐻𝑖𝑖[𝑓𝑓(π‘₯π‘₯)](𝑖𝑖 NOUN american_scientific_journal-3517 42 18 ) ) PUNCT american_scientific_journal-3517 42 19 = = PUNCT american_scientific_journal-3517 42 20 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– NUM american_scientific_journal-3517 42 21 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 42 22 2 2 NUM american_scientific_journal-3517 42 23 𝑑𝑑2 𝑑𝑑2 PROPN american_scientific_journal-3517 42 24 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 42 25 οΏ½ οΏ½ PROPN american_scientific_journal-3517 42 26 𝑓𝑓(π‘₯π‘₯ 𝑓𝑓(π‘₯π‘₯ PROPN american_scientific_journal-3517 42 27 ) ) PUNCT american_scientific_journal-3517 42 28 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 43 1 βˆ’ βˆ’ NOUN american_scientific_journal-3517 43 2 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-3517 43 3 ∞ ∞ PROPN american_scientific_journal-3517 43 4 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 43 5 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 43 6 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 43 7 2 2 NUM american_scientific_journal-3517 43 8 π‘₯π‘₯2𝑑𝑑π‘₯π‘₯ π‘₯π‘₯2𝑑𝑑π‘₯π‘₯ NOUN american_scientific_journal-3517 43 9 ( ( PUNCT american_scientific_journal-3517 43 10 2 2 X american_scientific_journal-3517 43 11 ) ) PUNCT american_scientific_journal-3517 43 12 𝐻𝐻𝑖𝑖 𝐻𝐻𝑖𝑖 ADJ american_scientific_journal-3517 43 13 οΏ½ οΏ½ VERB american_scientific_journal-3517 43 14 𝑒𝑒 𝑒𝑒 NOUN american_scientific_journal-3517 43 15 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 43 16 2(βˆ’π‘–π‘–2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 2(βˆ’π‘–π‘–2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 PROPN american_scientific_journal-3517 43 17 ) ) PUNCT american_scientific_journal-3517 43 18 οΏ½ οΏ½ PROPN american_scientific_journal-3517 43 19 ( ( PUNCT american_scientific_journal-3517 43 20 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 43 21 ) ) PUNCT american_scientific_journal-3517 43 22 = = NOUN american_scientific_journal-3517 44 1 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– NUM american_scientific_journal-3517 44 2 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 SYM american_scientific_journal-3517 44 3 2 2 NUM american_scientific_journal-3517 44 4 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 44 5 πœ‹πœ‹ πœ‹πœ‹ X american_scientific_journal-3517 44 6 οΏ½ οΏ½ PROPN american_scientific_journal-3517 44 7 𝑒𝑒(βˆ’π‘–π‘–2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑒𝑒(βˆ’π‘–π‘–2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 44 8 𝐹𝐹 𝐹𝐹 PROPN american_scientific_journal-3517 44 9 𝑖𝑖[𝑓𝑓(𝑖𝑖 𝑖𝑖[𝑓𝑓(𝑖𝑖 NUM american_scientific_journal-3517 44 10 ) ) PUNCT american_scientific_journal-3517 44 11 ] ] PUNCT american_scientific_journal-3517 45 1 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 45 2 βˆ’ βˆ’ PROPN american_scientific_journal-3517 45 3 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 45 4 ∞ ∞ PROPN american_scientific_journal-3517 45 5 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 45 6 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 45 7 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 45 8 2 2 NUM american_scientific_journal-3517 45 9 𝑖𝑖2𝑑𝑑𝑒𝑒 𝑖𝑖2𝑑𝑑𝑒𝑒 PUNCT american_scientific_journal-3517 45 10 = = NOUN american_scientific_journal-3517 45 11 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 45 12 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 SYM american_scientific_journal-3517 45 13 2 2 NUM american_scientific_journal-3517 45 14 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 45 15 πœ‹πœ‹ πœ‹πœ‹ X american_scientific_journal-3517 45 16 οΏ½ οΏ½ PROPN american_scientific_journal-3517 45 17 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– X american_scientific_journal-3517 45 18 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 45 19 2 2 NUM american_scientific_journal-3517 45 20 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 45 21 𝑦𝑦 𝑦𝑦 INTJ american_scientific_journal-3517 45 22 βˆ’ βˆ’ PROPN american_scientific_journal-3517 45 23 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 45 24 ∞ ∞ PROPN american_scientific_journal-3517 45 25 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 45 26 οΏ½ οΏ½ PROPN american_scientific_journal-3517 45 27 οΏ½ οΏ½ PROPN american_scientific_journal-3517 45 28 1 1 NUM american_scientific_journal-3517 45 29 βˆ’ βˆ’ PROPN american_scientific_journal-3517 45 30 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 45 31 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 45 32 οΏ½ οΏ½ PROPN american_scientific_journal-3517 46 1 𝑓𝑓(𝑒𝑒𝑑𝑑 𝑓𝑓(𝑒𝑒𝑑𝑑 PRON american_scientific_journal-3517 46 2 ) ) PUNCT american_scientific_journal-3517 47 1 ∞ ∞ PROPN american_scientific_journal-3517 47 2 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 47 3 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 47 4 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 47 5 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 47 6 οΏ½ οΏ½ NOUN american_scientific_journal-3517 47 7 𝑑𝑑𝑒𝑒 𝑑𝑑𝑒𝑒 ADJ american_scientific_journal-3517 47 8 = = SYM american_scientific_journal-3517 47 9 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 47 10 𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐2 𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐2 NOUN american_scientific_journal-3517 47 11 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 47 12 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 47 13 οΏ½ οΏ½ PROPN american_scientific_journal-3517 47 14 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– NUM american_scientific_journal-3517 47 15 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 47 16 ∫ ∫ PROPN american_scientific_journal-3517 47 17 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 PROPN american_scientific_journal-3517 47 18 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 PROPN american_scientific_journal-3517 47 19 2 2 NUM american_scientific_journal-3517 47 20 𝑐𝑐2 𝑐𝑐2 NOUN american_scientific_journal-3517 47 21 π‘¦π‘¦βˆ’π‘–π‘– π‘¦π‘¦βˆ’π‘–π‘– NOUN american_scientific_journal-3517 47 22 ∞ ∞ PROPN american_scientific_journal-3517 47 23 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 47 24 οΏ½ οΏ½ PROPN american_scientific_journal-3517 47 25 ∫ ∫ PROPN american_scientific_journal-3517 47 26 𝑓𝑓(𝑒𝑒𝑑𝑑)∞ 𝑓𝑓(𝑒𝑒𝑑𝑑)∞ NOUN american_scientific_journal-3517 47 27 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 47 28 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– NOUN american_scientific_journal-3517 47 29 οΏ½ οΏ½ NOUN american_scientific_journal-3517 47 30 𝑑𝑑𝑒𝑒 𝑑𝑑𝑒𝑒 VERB american_scientific_journal-3517 47 31 by by ADP american_scientific_journal-3517 47 32 changing change VERB american_scientific_journal-3517 47 33 the the DET american_scientific_journal-3517 47 34 order order NOUN american_scientific_journal-3517 47 35 of of ADP american_scientific_journal-3517 47 36 integration integration NOUN american_scientific_journal-3517 47 37 = = NOUN american_scientific_journal-3517 47 38 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 47 39 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 47 40 2 2 NUM american_scientific_journal-3517 47 41 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 47 42 οΏ½ οΏ½ NOUN american_scientific_journal-3517 47 43 1 1 NUM american_scientific_journal-3517 47 44 βˆ’ βˆ’ PROPN american_scientific_journal-3517 47 45 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 47 46 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 47 47 οΏ½ οΏ½ PROPN american_scientific_journal-3517 47 48 𝑓𝑓(𝑒𝑒𝑑𝑑)𝑒𝑒𝑖𝑖 𝑓𝑓(𝑒𝑒𝑑𝑑)𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 47 49 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 47 50 2 2 NUM american_scientific_journal-3517 47 51 𝑑𝑑2 𝑑𝑑2 NOUN american_scientific_journal-3517 47 52 ∞ ∞ PROPN american_scientific_journal-3517 48 1 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 48 2 οΏ½ οΏ½ PROPN american_scientific_journal-3517 48 3 1 1 NUM american_scientific_journal-3517 48 4 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 48 5 οΏ½ οΏ½ PROPN american_scientific_journal-3517 48 6 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘– π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘– PROPN american_scientific_journal-3517 48 7 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 48 8 βˆ’ βˆ’ PROPN american_scientific_journal-3517 48 9 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 48 10 ∞ ∞ PROPN american_scientific_journal-3517 48 11 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 48 12 𝑑𝑑𝑒𝑒 𝑑𝑑𝑒𝑒 PROPN american_scientific_journal-3517 48 13 οΏ½ οΏ½ PROPN american_scientific_journal-3517 48 14 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 48 15 ( ( PUNCT american_scientific_journal-3517 48 16 3 3 NUM american_scientific_journal-3517 48 17 ) ) PUNCT american_scientific_journal-3517 48 18 from from ADP american_scientific_journal-3517 48 19 [ [ X american_scientific_journal-3517 48 20 18 18 NUM american_scientific_journal-3517 48 21 ] ] X american_scientific_journal-3517 48 22 we we PRON american_scientific_journal-3517 48 23 have have VERB american_scientific_journal-3517 48 24 οΏ½ οΏ½ PROPN american_scientific_journal-3517 48 25 𝑒𝑒𝑖𝑖𝑖𝑖𝑑𝑑 𝑒𝑒𝑖𝑖𝑖𝑖𝑑𝑑 ADJ american_scientific_journal-3517 48 26 π‘₯π‘₯ π‘₯π‘₯ ADP american_scientific_journal-3517 48 27 βˆ’ βˆ’ PROPN american_scientific_journal-3517 48 28 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 48 29 ∞ ∞ PROPN american_scientific_journal-3517 48 30 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 48 31 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 48 32 = = SYM american_scientific_journal-3517 48 33 βˆ’π‘–π‘– βˆ’π‘–π‘– NOUN american_scientific_journal-3517 48 34 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑖𝑖π‘₯π‘₯ 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑖𝑖π‘₯π‘₯ PROPN american_scientific_journal-3517 48 35 ( ( PUNCT american_scientific_journal-3517 48 36 4 4 NUM american_scientific_journal-3517 48 37 ) ) PUNCT american_scientific_journal-3517 48 38 equation equation NOUN american_scientific_journal-3517 48 39 ( ( PUNCT american_scientific_journal-3517 48 40 3 3 X american_scientific_journal-3517 48 41 ) ) PUNCT american_scientific_journal-3517 48 42 can can AUX american_scientific_journal-3517 48 43 be be AUX american_scientific_journal-3517 48 44 written write VERB american_scientific_journal-3517 48 45 as as ADP american_scientific_journal-3517 48 46 = = NOUN american_scientific_journal-3517 48 47 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 48 48 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 48 49 2 2 NUM american_scientific_journal-3517 48 50 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 48 51 οΏ½ οΏ½ NOUN american_scientific_journal-3517 48 52 1 1 NUM american_scientific_journal-3517 48 53 βˆ’ βˆ’ PROPN american_scientific_journal-3517 48 54 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 48 55 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 48 56 οΏ½ οΏ½ PROPN american_scientific_journal-3517 49 1 𝑓𝑓(𝑒𝑒𝑑𝑑 𝑓𝑓(𝑒𝑒𝑑𝑑 PRON american_scientific_journal-3517 49 2 ) ) PUNCT american_scientific_journal-3517 49 3 ∞ ∞ PROPN american_scientific_journal-3517 49 4 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 49 5 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 49 6 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 49 7 2 2 NUM american_scientific_journal-3517 49 8 𝑑𝑑2(𝑖𝑖)𝑠𝑠𝑠𝑠𝑠𝑠 𝑑𝑑2(𝑖𝑖)𝑠𝑠𝑠𝑠𝑠𝑠 ADP american_scientific_journal-3517 49 9 οΏ½ οΏ½ PROPN american_scientific_journal-3517 49 10 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 49 11 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 NOUN american_scientific_journal-3517 49 12 οΏ½ οΏ½ PROPN american_scientific_journal-3517 49 13 π‘’π‘’βˆ’π‘–π‘–π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– π‘’π‘’βˆ’π‘–π‘–π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– PROPN american_scientific_journal-3517 49 14 = = NOUN american_scientific_journal-3517 49 15 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 49 16 οΏ½ οΏ½ NOUN american_scientific_journal-3517 49 17 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– NUM american_scientific_journal-3517 49 18 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 49 19 ∫ ∫ PROPN american_scientific_journal-3517 50 1 𝑓𝑓(𝑒𝑒𝑑𝑑)∞ 𝑓𝑓(𝑒𝑒𝑑𝑑)∞ NOUN american_scientific_journal-3517 50 2 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 50 3 𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠 NOUN american_scientific_journal-3517 50 4 οΏ½ οΏ½ PROPN american_scientific_journal-3517 50 5 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-3517 50 6 𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖 𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖 PROPN american_scientific_journal-3517 50 7 οΏ½ οΏ½ PROPN american_scientific_journal-3517 50 8 π‘’π‘’βˆ’π‘–π‘–π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 50 9 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 50 10 2(𝑑𝑑2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2(𝑑𝑑2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 50 11 𝐻𝐻𝑖𝑖{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)]}(𝑒𝑒 𝐻𝐻𝑖𝑖{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)]}(𝑒𝑒 PROPN american_scientific_journal-3517 50 12 ) ) PUNCT american_scientific_journal-3517 50 13 = = PUNCT american_scientific_journal-3517 50 14 𝑖𝑖𝐹𝐹𝑖𝑖 𝑖𝑖𝐹𝐹𝑖𝑖 PUNCT american_scientific_journal-3517 50 15 οΏ½ οΏ½ PROPN american_scientific_journal-3517 50 16 𝑆𝑆𝑠𝑠𝑠𝑠 𝑆𝑆𝑠𝑠𝑠𝑠 PROPN american_scientific_journal-3517 50 17 οΏ½ οΏ½ PROPN american_scientific_journal-3517 50 18 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 50 19 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 NOUN american_scientific_journal-3517 50 20 οΏ½ οΏ½ PROPN american_scientific_journal-3517 50 21 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 50 22 ) ) PUNCT american_scientific_journal-3517 50 23 οΏ½ οΏ½ PROPN american_scientific_journal-3517 50 24 ( ( PUNCT american_scientific_journal-3517 50 25 5 5 NUM american_scientific_journal-3517 50 26 ) ) PUNCT american_scientific_journal-3517 50 27 equation equation NOUN american_scientific_journal-3517 50 28 ( ( PUNCT american_scientific_journal-3517 50 29 4 4 NUM american_scientific_journal-3517 50 30 ) ) PUNCT american_scientific_journal-3517 50 31 is be AUX american_scientific_journal-3517 50 32 the the DET american_scientific_journal-3517 50 33 relation relation NOUN american_scientific_journal-3517 50 34 between between ADP american_scientific_journal-3517 50 35 generalized generalized ADJ american_scientific_journal-3517 50 36 fractional fractional ADJ american_scientific_journal-3517 50 37 hilbert hilbert NOUN american_scientific_journal-3517 50 38 transform transform NOUN american_scientific_journal-3517 50 39 and and CCONJ american_scientific_journal-3517 50 40 fractional fractional ADJ american_scientific_journal-3517 50 41 mellin mellin PROPN american_scientific_journal-3517 50 42 transform transform NOUN american_scientific_journal-3517 50 43 . . PUNCT american_scientific_journal-3517 51 1 american american ADJ american_scientific_journal-3517 51 2 scientific scientific ADJ american_scientific_journal-3517 51 3 research research PROPN american_scientific_journal-3517 51 4 journal journal NOUN american_scientific_journal-3517 51 5 for for ADP american_scientific_journal-3517 51 6 engineering engineering NOUN american_scientific_journal-3517 51 7 , , PUNCT american_scientific_journal-3517 51 8 technology technology NOUN american_scientific_journal-3517 51 9 , , PUNCT american_scientific_journal-3517 51 10 and and CCONJ american_scientific_journal-3517 51 11 sciences science NOUN american_scientific_journal-3517 51 12 ( ( PUNCT american_scientific_journal-3517 51 13 asrjets asrjet NOUN american_scientific_journal-3517 51 14 ) ) PUNCT american_scientific_journal-3517 51 15 ( ( PUNCT american_scientific_journal-3517 51 16 2017 2017 NUM american_scientific_journal-3517 51 17 ) ) PUNCT american_scientific_journal-3517 51 18 volume volume NOUN american_scientific_journal-3517 51 19 38 38 NUM american_scientific_journal-3517 51 20 , , PUNCT american_scientific_journal-3517 51 21 no no DET american_scientific_journal-3517 51 22 2 2 NUM american_scientific_journal-3517 51 23 , , PUNCT american_scientific_journal-3517 51 24 pp pp ADV american_scientific_journal-3517 51 25 90 90 NUM american_scientific_journal-3517 51 26 - - SYM american_scientific_journal-3517 51 27 97 97 NUM american_scientific_journal-3517 51 28 93 93 NUM american_scientific_journal-3517 51 29 2.2 2.2 NUM american_scientific_journal-3517 51 30 . . PUNCT american_scientific_journal-3517 52 1 the the DET american_scientific_journal-3517 52 2 relationship relationship NOUN american_scientific_journal-3517 52 3 between between ADP american_scientific_journal-3517 52 4 generalized generalized ADJ american_scientific_journal-3517 52 5 fractional fractional ADJ american_scientific_journal-3517 52 6 inverse inverse NOUN american_scientific_journal-3517 52 7 hilbert hilbert NOUN american_scientific_journal-3517 52 8 transform transform NOUN american_scientific_journal-3517 52 9 with with ADP american_scientific_journal-3517 52 10 fractional fractional ADJ american_scientific_journal-3517 52 11 mellin mellin PROPN american_scientific_journal-3517 52 12 transform transform NOUN american_scientific_journal-3517 52 13 let let VERB american_scientific_journal-3517 52 14 𝑠𝑠(𝑒𝑒 𝑠𝑠(𝑒𝑒 PROPN american_scientific_journal-3517 52 15 ) ) PUNCT american_scientific_journal-3517 53 1 = = SYM american_scientific_journal-3517 53 2 ∫ ∫ PROPN american_scientific_journal-3517 54 1 𝑓𝑓(𝑖𝑖)∞ 𝑓𝑓(𝑖𝑖)∞ PROPN american_scientific_journal-3517 54 2 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 54 3 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 54 4 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 54 5 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 54 6 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 54 7 ( ( PUNCT american_scientific_journal-3517 54 8 5 5 NUM american_scientific_journal-3517 54 9 ) ) PUNCT american_scientific_journal-3517 54 10 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 54 11 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 54 12 2 2 NUM american_scientific_journal-3517 54 13 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 54 14 οΏ½ οΏ½ PROPN american_scientific_journal-3517 54 15 1 1 NUM american_scientific_journal-3517 54 16 βˆ’ βˆ’ PROPN american_scientific_journal-3517 54 17 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 54 18 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 54 19 𝑠𝑠(𝑒𝑒 𝑠𝑠(𝑒𝑒 PROPN american_scientific_journal-3517 54 20 ) ) PUNCT american_scientific_journal-3517 54 21 = = SYM american_scientific_journal-3517 54 22 οΏ½ οΏ½ PROPN american_scientific_journal-3517 54 23 1 1 NUM american_scientific_journal-3517 54 24 βˆ’ βˆ’ PROPN american_scientific_journal-3517 54 25 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 54 26 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 54 27 οΏ½ οΏ½ PROPN american_scientific_journal-3517 54 28 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 54 29 ) ) PUNCT american_scientific_journal-3517 54 30 ∞ ∞ PROPN american_scientific_journal-3517 54 31 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 54 32 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 54 33 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 54 34 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 54 35 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 54 36 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 PROPN american_scientific_journal-3517 54 37 2 2 NUM american_scientific_journal-3517 54 38 𝑖𝑖2 𝑖𝑖2 PROPN american_scientific_journal-3517 54 39 οΏ½ οΏ½ PROPN american_scientific_journal-3517 54 40 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– NUM american_scientific_journal-3517 54 41 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 54 42 𝑠𝑠(𝑒𝑒 𝑠𝑠(𝑒𝑒 PROPN american_scientific_journal-3517 54 43 ) ) PUNCT american_scientific_journal-3517 54 44 = = SYM american_scientific_journal-3517 54 45 𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 NOUN american_scientific_journal-3517 54 46 ) ) PUNCT american_scientific_journal-3517 54 47 ( ( PUNCT american_scientific_journal-3517 54 48 6 6 NUM american_scientific_journal-3517 54 49 ) ) PUNCT american_scientific_journal-3517 54 50 π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦 π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦 NOUN american_scientific_journal-3517 54 51 ) ) PUNCT american_scientific_journal-3517 54 52 = = SYM american_scientific_journal-3517 54 53 βˆ’ βˆ’ NOUN american_scientific_journal-3517 54 54 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 54 55 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 54 56 2 2 NUM american_scientific_journal-3517 54 57 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 54 58 πœ‹πœ‹ πœ‹πœ‹ X american_scientific_journal-3517 54 59 οΏ½ οΏ½ PROPN american_scientific_journal-3517 54 60 𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 NOUN american_scientific_journal-3517 54 61 ) ) PUNCT american_scientific_journal-3517 54 62 𝑦𝑦 𝑦𝑦 VERB american_scientific_journal-3517 54 63 βˆ’ βˆ’ PROPN american_scientific_journal-3517 54 64 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 54 65 ∞ ∞ NUM american_scientific_journal-3517 54 66 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 54 67 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 54 68 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 54 69 2 2 NUM american_scientific_journal-3517 54 70 𝑖𝑖2𝑑𝑑𝑒𝑒 𝑖𝑖2𝑑𝑑𝑒𝑒 PUNCT american_scientific_journal-3517 55 1 = = NOUN american_scientific_journal-3517 55 2 βˆ’π‘’π‘’π‘–π‘– βˆ’π‘’π‘’π‘–π‘– NOUN american_scientific_journal-3517 55 3 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 ADJ american_scientific_journal-3517 55 4 2 2 NUM american_scientific_journal-3517 55 5 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 55 6 πœ‹πœ‹ πœ‹πœ‹ PUNCT american_scientific_journal-3517 55 7 οΏ½ οΏ½ PROPN american_scientific_journal-3517 55 8 1 1 NUM american_scientific_journal-3517 55 9 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 55 10 βˆ’ βˆ’ PROPN american_scientific_journal-3517 55 11 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 55 12 ∞ ∞ NUM american_scientific_journal-3517 55 13 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 55 14 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 55 15 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 55 16 2 2 NUM american_scientific_journal-3517 55 17 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 55 18 οΏ½ οΏ½ PROPN american_scientific_journal-3517 55 19 1 1 NUM american_scientific_journal-3517 55 20 βˆ’ βˆ’ PROPN american_scientific_journal-3517 55 21 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 55 22 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 55 23 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 55 24 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 55 25 2 2 NUM american_scientific_journal-3517 55 26 𝑖𝑖2𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 𝑖𝑖2𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 VERB american_scientific_journal-3517 55 27 = = NOUN american_scientific_journal-3517 55 28 βˆ’π‘’π‘’π‘–π‘– βˆ’π‘’π‘’π‘–π‘– NOUN american_scientific_journal-3517 55 29 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 55 30 2 2 NUM american_scientific_journal-3517 55 31 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 55 32 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-3517 55 33 οΏ½ οΏ½ PROPN american_scientific_journal-3517 55 34 1 1 NUM american_scientific_journal-3517 55 35 βˆ’ βˆ’ PROPN american_scientific_journal-3517 55 36 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 55 37 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 55 38 οΏ½ οΏ½ PROPN american_scientific_journal-3517 55 39 1 1 NUM american_scientific_journal-3517 55 40 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 55 41 βˆ’ βˆ’ PROPN american_scientific_journal-3517 55 42 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 55 43 ∞ ∞ PROPN american_scientific_journal-3517 55 44 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 55 45 οΏ½ οΏ½ PROPN american_scientific_journal-3517 55 46 οΏ½ οΏ½ PROPN american_scientific_journal-3517 55 47 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 55 48 ) ) PUNCT american_scientific_journal-3517 55 49 ∞ ∞ PROPN american_scientific_journal-3517 55 50 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 55 51 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 55 52 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 55 53 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 55 54 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 55 55 οΏ½ οΏ½ PROPN american_scientific_journal-3517 55 56 𝑑𝑑𝑒𝑒 𝑑𝑑𝑒𝑒 VERB american_scientific_journal-3517 55 57 𝑏𝑏𝑦𝑦 𝑏𝑏𝑦𝑦 NOUN american_scientific_journal-3517 55 58 π‘–π‘–β„Žπ‘Žπ‘Žπ‘ π‘ π‘ π‘ π‘–π‘–π‘ π‘ π‘ π‘  π‘–π‘–β„Žπ‘Žπ‘Žπ‘ π‘ π‘ π‘ π‘–π‘–π‘ π‘ π‘ π‘  PROPN american_scientific_journal-3517 55 59 π‘–π‘–β„Žπ‘’π‘’ π‘–π‘–β„Žπ‘’π‘’ PROPN american_scientific_journal-3517 55 60 π‘–π‘–π‘œπ‘œπ‘‘π‘‘π‘’π‘’π‘œπ‘œ π‘–π‘–π‘œπ‘œπ‘‘π‘‘π‘’π‘’π‘œπ‘œ PROPN american_scientific_journal-3517 55 61 𝑖𝑖𝑓𝑓 𝑖𝑖𝑓𝑓 PROPN american_scientific_journal-3517 56 1 π‘–π‘–π‘ π‘ π‘–π‘–π‘’π‘’π‘ π‘ π‘œπ‘œπ‘Žπ‘Žπ‘–π‘–π‘–π‘–π‘–π‘–π‘ π‘  π‘–π‘–π‘ π‘ π‘–π‘–π‘’π‘’π‘ π‘ π‘œπ‘œπ‘Žπ‘Žπ‘–π‘–π‘–π‘–π‘–π‘–π‘ π‘  PROPN american_scientific_journal-3517 56 2 π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦 π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦 NOUN american_scientific_journal-3517 56 3 ) ) PUNCT american_scientific_journal-3517 57 1 = = PRON american_scientific_journal-3517 57 2 βˆ’π‘’π‘’π‘–π‘– βˆ’π‘’π‘’π‘–π‘– NOUN american_scientific_journal-3517 57 3 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 ADJ american_scientific_journal-3517 57 4 2 2 NUM american_scientific_journal-3517 57 5 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 57 6 πœ‹πœ‹ πœ‹πœ‹ NOUN american_scientific_journal-3517 57 7 οΏ½ οΏ½ PROPN american_scientific_journal-3517 57 8 1 1 NUM american_scientific_journal-3517 57 9 βˆ’ βˆ’ PROPN american_scientific_journal-3517 57 10 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 57 11 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 57 12 οΏ½ οΏ½ PROPN american_scientific_journal-3517 57 13 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 57 14 ) ) PUNCT american_scientific_journal-3517 57 15 ∞ ∞ PROPN american_scientific_journal-3517 57 16 βˆ’βˆž βˆ’βˆž X american_scientific_journal-3517 57 17 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 57 18 𝑖𝑖 𝑖𝑖 NUM american_scientific_journal-3517 57 19 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 57 20 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 57 21 οΏ½ οΏ½ PROPN american_scientific_journal-3517 57 22 1 1 NUM american_scientific_journal-3517 57 23 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 57 24 οΏ½ οΏ½ PROPN american_scientific_journal-3517 57 25 1 1 NUM american_scientific_journal-3517 57 26 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 57 27 βˆ’ βˆ’ PROPN american_scientific_journal-3517 57 28 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 57 29 ∞ ∞ NUM american_scientific_journal-3517 57 30 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 57 31 π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘’π‘’ NUM american_scientific_journal-3517 57 32 οΏ½ οΏ½ PROPN american_scientific_journal-3517 57 33 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 57 34 ( ( PUNCT american_scientific_journal-3517 57 35 7 7 X american_scientific_journal-3517 57 36 ) ) PUNCT american_scientific_journal-3517 57 37 we we PRON american_scientific_journal-3517 57 38 have have VERB american_scientific_journal-3517 57 39 from from ADP american_scientific_journal-3517 57 40 [ [ X american_scientific_journal-3517 57 41 18 18 NUM american_scientific_journal-3517 57 42 ] ] PUNCT american_scientific_journal-3517 57 43 ∫ ∫ PROPN american_scientific_journal-3517 57 44 𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐 PROPN american_scientific_journal-3517 57 45 π‘₯π‘₯βˆ’π‘‘π‘‘ π‘₯π‘₯βˆ’π‘‘π‘‘ PROPN american_scientific_journal-3517 57 46 ∞ ∞ PROPN american_scientific_journal-3517 57 47 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 57 48 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 57 49 = = SYM american_scientific_journal-3517 57 50 βˆ’π‘–π‘– βˆ’π‘–π‘– NOUN american_scientific_journal-3517 57 51 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑖𝑖π‘₯π‘₯ 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑖𝑖π‘₯π‘₯ PROPN american_scientific_journal-3517 57 52 ( ( PUNCT american_scientific_journal-3517 57 53 8) 8) NUM american_scientific_journal-3517 57 54 equation equation NOUN american_scientific_journal-3517 57 55 ( ( PUNCT american_scientific_journal-3517 57 56 7 7 X american_scientific_journal-3517 57 57 ) ) PUNCT american_scientific_journal-3517 57 58 can can AUX american_scientific_journal-3517 57 59 be be AUX american_scientific_journal-3517 57 60 written write VERB american_scientific_journal-3517 57 61 as as ADP american_scientific_journal-3517 57 62 = = PUNCT american_scientific_journal-3517 57 63 βˆ’π‘–π‘–π‘–π‘– βˆ’π‘–π‘–π‘–π‘– NOUN american_scientific_journal-3517 57 64 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 NOUN american_scientific_journal-3517 57 65 2 2 NUM american_scientific_journal-3517 57 66 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 57 67 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 57 68 οΏ½ οΏ½ PROPN american_scientific_journal-3517 57 69 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– NUM american_scientific_journal-3517 57 70 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 57 71 ∫ ∫ PROPN american_scientific_journal-3517 58 1 𝑓𝑓(𝑖𝑖)∞ 𝑓𝑓(𝑖𝑖)∞ PROPN american_scientific_journal-3517 58 2 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 58 3 𝑒𝑒 𝑒𝑒 X american_scientific_journal-3517 58 4 𝑖𝑖 𝑖𝑖 X american_scientific_journal-3517 58 5 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 58 6 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 58 7 . . PUNCT american_scientific_journal-3517 59 1 𝑖𝑖. 𝑖𝑖. ADV american_scientific_journal-3517 59 2 𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠 VERB american_scientific_journal-3517 59 3 οΏ½ οΏ½ PROPN american_scientific_journal-3517 59 4 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-3517 59 5 𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖 𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖 ADJ american_scientific_journal-3517 59 6 οΏ½ οΏ½ PROPN american_scientific_journal-3517 59 7 π‘’π‘’βˆ’π‘–π‘–π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– π‘’π‘’βˆ’π‘–π‘–π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– PROPN american_scientific_journal-3517 59 8 π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦 π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦 NOUN american_scientific_journal-3517 59 9 ) ) PUNCT american_scientific_journal-3517 60 1 = = SYM american_scientific_journal-3517 60 2 βˆ’π‘–π‘–πΉπΉπ‘–π‘– βˆ’π‘–π‘–πΉπΉπ‘–π‘– PROPN american_scientific_journal-3517 60 3 οΏ½ οΏ½ PROPN american_scientific_journal-3517 60 4 𝑆𝑆𝑠𝑠𝑠𝑠 𝑆𝑆𝑠𝑠𝑠𝑠 PROPN american_scientific_journal-3517 60 5 οΏ½ οΏ½ PROPN american_scientific_journal-3517 60 6 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 60 7 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 NOUN american_scientific_journal-3517 60 8 οΏ½ οΏ½ PROPN american_scientific_journal-3517 60 9 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 60 10 ) ) PUNCT american_scientific_journal-3517 60 11 οΏ½ οΏ½ PROPN american_scientific_journal-3517 60 12 ( ( PUNCT american_scientific_journal-3517 60 13 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 60 14 ) ) PUNCT american_scientific_journal-3517 60 15 ( ( PUNCT american_scientific_journal-3517 60 16 9 9 X american_scientific_journal-3517 60 17 ) ) PUNCT american_scientific_journal-3517 60 18 equation equation NOUN american_scientific_journal-3517 60 19 ( ( PUNCT american_scientific_journal-3517 60 20 9 9 NUM american_scientific_journal-3517 60 21 ) ) PUNCT american_scientific_journal-3517 60 22 is be AUX american_scientific_journal-3517 60 23 the the DET american_scientific_journal-3517 60 24 relation relation NOUN american_scientific_journal-3517 60 25 between between ADP american_scientific_journal-3517 60 26 generalized generalized ADJ american_scientific_journal-3517 60 27 fractional fractional ADJ american_scientific_journal-3517 60 28 inverse inverse NOUN american_scientific_journal-3517 60 29 hilbert hilbert NOUN american_scientific_journal-3517 60 30 transform transform NOUN american_scientific_journal-3517 60 31 and and CCONJ american_scientific_journal-3517 60 32 fractional fractional ADJ american_scientific_journal-3517 60 33 mellin mellin PROPN american_scientific_journal-3517 60 34 transform transform NOUN american_scientific_journal-3517 60 35 . . PUNCT american_scientific_journal-3517 61 1 2.3 2.3 NUM american_scientific_journal-3517 61 2 relationship relationship NOUN american_scientific_journal-3517 61 3 between between ADP american_scientific_journal-3517 61 4 generalized generalized ADJ american_scientific_journal-3517 61 5 fractional fractional ADJ american_scientific_journal-3517 61 6 invers inver NOUN american_scientific_journal-3517 61 7 hilbert hilbert PROPN american_scientific_journal-3517 61 8 transform transform VERB american_scientific_journal-3517 61 9 with with ADP american_scientific_journal-3517 61 10 fractional fractional ADJ american_scientific_journal-3517 61 11 laplace laplace NOUN american_scientific_journal-3517 61 12 transform transform VERB american_scientific_journal-3517 61 13 american american ADJ american_scientific_journal-3517 61 14 scientific scientific ADJ american_scientific_journal-3517 61 15 research research NOUN american_scientific_journal-3517 61 16 journal journal NOUN american_scientific_journal-3517 61 17 for for ADP american_scientific_journal-3517 61 18 engineering engineering NOUN american_scientific_journal-3517 61 19 , , PUNCT american_scientific_journal-3517 61 20 technology technology NOUN american_scientific_journal-3517 61 21 , , PUNCT american_scientific_journal-3517 61 22 and and CCONJ american_scientific_journal-3517 61 23 sciences science NOUN american_scientific_journal-3517 61 24 ( ( PUNCT american_scientific_journal-3517 61 25 asrjets asrjet NOUN american_scientific_journal-3517 61 26 ) ) PUNCT american_scientific_journal-3517 61 27 ( ( PUNCT american_scientific_journal-3517 61 28 2017 2017 NUM american_scientific_journal-3517 61 29 ) ) PUNCT american_scientific_journal-3517 61 30 volume volume NOUN american_scientific_journal-3517 61 31 38 38 NUM american_scientific_journal-3517 61 32 , , PUNCT american_scientific_journal-3517 61 33 no no DET american_scientific_journal-3517 61 34 2 2 NUM american_scientific_journal-3517 61 35 , , PUNCT american_scientific_journal-3517 61 36 pp pp ADV american_scientific_journal-3517 61 37 90 90 NUM american_scientific_journal-3517 61 38 - - SYM american_scientific_journal-3517 61 39 97 97 NUM american_scientific_journal-3517 61 40 94 94 NUM american_scientific_journal-3517 61 41 fractional fractional ADJ american_scientific_journal-3517 61 42 laplace laplace NOUN american_scientific_journal-3517 61 43 transform transform NOUN american_scientific_journal-3517 61 44 is be AUX american_scientific_journal-3517 61 45 defined define VERB american_scientific_journal-3517 61 46 in in ADP american_scientific_journal-3517 61 47 [ [ X american_scientific_journal-3517 61 48 24 24 NUM american_scientific_journal-3517 61 49 ] ] PUNCT american_scientific_journal-3517 61 50 as as ADP american_scientific_journal-3517 61 51 𝐿𝐿𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 𝐿𝐿𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 NOUN american_scientific_journal-3517 61 52 ) ) PUNCT american_scientific_journal-3517 61 53 = = PUNCT american_scientific_journal-3517 61 54 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– NUM american_scientific_journal-3517 61 55 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 61 56 2 2 NUM american_scientific_journal-3517 61 57 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 61 58 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 59 1 1 NUM american_scientific_journal-3517 61 60 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 61 61 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 62 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 61 63 ) ) PUNCT american_scientific_journal-3517 61 64 ∞ ∞ PROPN american_scientific_journal-3517 61 65 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 61 66 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 61 67 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 61 68 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 61 69 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 61 70 ( ( PUNCT american_scientific_journal-3517 61 71 10 10 NUM american_scientific_journal-3517 61 72 ) ) PUNCT american_scientific_journal-3517 61 73 using use VERB american_scientific_journal-3517 61 74 equation equation NOUN american_scientific_journal-3517 61 75 ( ( PUNCT american_scientific_journal-3517 61 76 5 5 NUM american_scientific_journal-3517 61 77 ) ) PUNCT american_scientific_journal-3517 61 78 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 61 79 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 61 80 2 2 NUM american_scientific_journal-3517 61 81 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 61 82 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 83 1 1 NUM american_scientific_journal-3517 61 84 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 61 85 𝑠𝑠(𝑒𝑒 𝑠𝑠(𝑒𝑒 NUM american_scientific_journal-3517 61 86 ) ) PUNCT american_scientific_journal-3517 61 87 = = PUNCT american_scientific_journal-3517 61 88 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– NUM american_scientific_journal-3517 61 89 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 SYM american_scientific_journal-3517 61 90 2 2 NUM american_scientific_journal-3517 61 91 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 61 92 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 93 1 1 NUM american_scientific_journal-3517 61 94 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 61 95 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 96 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 61 97 ) ) PUNCT american_scientific_journal-3517 61 98 ∞ ∞ PROPN american_scientific_journal-3517 61 99 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 61 100 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 61 101 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 61 102 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 61 103 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 61 104 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 61 105 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 61 106 2 2 NUM american_scientific_journal-3517 61 107 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 61 108 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 109 1 1 NUM american_scientific_journal-3517 61 110 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 61 111 𝑠𝑠(𝑒𝑒 𝑠𝑠(𝑒𝑒 NUM american_scientific_journal-3517 61 112 ) ) PUNCT american_scientific_journal-3517 61 113 = = SYM american_scientific_journal-3517 61 114 𝐿𝐿𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 𝐿𝐿𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒 NOUN american_scientific_journal-3517 61 115 ) ) PUNCT american_scientific_journal-3517 61 116 ( ( PUNCT american_scientific_journal-3517 61 117 11 11 NUM american_scientific_journal-3517 61 118 ) ) PUNCT american_scientific_journal-3517 61 119 now now ADV american_scientific_journal-3517 61 120 consider consider VERB american_scientific_journal-3517 61 121 π»π»βˆ’π‘–π‘– π»π»βˆ’π‘–π‘– PROPN american_scientific_journal-3517 61 122 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 123 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖(𝑖𝑖2βˆ’π‘¦π‘¦2)𝐿𝐿𝑖𝑖[𝐹𝐹(𝑖𝑖)](𝑒𝑒) 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖(𝑖𝑖2βˆ’π‘¦π‘¦2)𝐿𝐿𝑖𝑖[𝐹𝐹(𝑖𝑖)](𝑒𝑒) PROPN american_scientific_journal-3517 61 124 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 125 (𝑦𝑦 (𝑦𝑦 NOUN american_scientific_journal-3517 61 126 ) ) PUNCT american_scientific_journal-3517 61 127 = = SYM american_scientific_journal-3517 61 128 βˆ’ βˆ’ PROPN american_scientific_journal-3517 61 129 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 61 130 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 61 131 2 2 NUM american_scientific_journal-3517 61 132 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 61 133 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 61 134 οΏ½ οΏ½ PROPN american_scientific_journal-3517 61 135 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖(𝑖𝑖2βˆ’π‘¦π‘¦2 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖(𝑖𝑖2βˆ’π‘¦π‘¦2 PROPN american_scientific_journal-3517 61 136 ) ) PUNCT american_scientific_journal-3517 61 137 𝐿𝐿 𝐿𝐿 PROPN american_scientific_journal-3517 61 138 𝑖𝑖[𝐹𝐹(𝑖𝑖)](𝑒𝑒 𝑖𝑖[𝐹𝐹(𝑖𝑖)](𝑒𝑒 PROPN american_scientific_journal-3517 61 139 ) ) PUNCT american_scientific_journal-3517 61 140 𝑦𝑦 𝑦𝑦 ADP american_scientific_journal-3517 61 141 βˆ’ βˆ’ PROPN american_scientific_journal-3517 61 142 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 61 143 ∞ ∞ NUM american_scientific_journal-3517 61 144 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 61 145 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 61 146 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 61 147 2 2 NUM american_scientific_journal-3517 61 148 𝑖𝑖2𝑑𝑑𝑒𝑒 𝑖𝑖2𝑑𝑑𝑒𝑒 PUNCT american_scientific_journal-3517 62 1 = = NOUN american_scientific_journal-3517 62 2 βˆ’ βˆ’ PROPN american_scientific_journal-3517 62 3 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– NUM american_scientific_journal-3517 62 4 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 62 5 2 2 NUM american_scientific_journal-3517 62 6 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 62 7 πœ‹πœ‹ πœ‹πœ‹ PUNCT american_scientific_journal-3517 62 8 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 9 1 1 NUM american_scientific_journal-3517 62 10 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 62 11 βˆ’ βˆ’ PROPN american_scientific_journal-3517 62 12 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 62 13 ∞ ∞ PROPN american_scientific_journal-3517 62 14 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 62 15 𝑒𝑒𝑖𝑖 𝑒𝑒𝑖𝑖 NOUN american_scientific_journal-3517 62 16 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 62 17 2 2 NUM american_scientific_journal-3517 62 18 𝑖𝑖2π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖2π‘’π‘’βˆ’π‘–π‘– NOUN american_scientific_journal-3517 62 19 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 ADP american_scientific_journal-3517 62 20 2 2 NUM american_scientific_journal-3517 62 21 𝑖𝑖2 𝑖𝑖2 NOUN american_scientific_journal-3517 62 22 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 23 1 1 NUM american_scientific_journal-3517 62 24 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 62 25 𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 VERB american_scientific_journal-3517 62 26 = = SYM american_scientific_journal-3517 62 27 βˆ’ βˆ’ NOUN american_scientific_journal-3517 62 28 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 62 29 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 62 30 2 2 NUM american_scientific_journal-3517 62 31 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 62 32 πœ‹πœ‹ πœ‹πœ‹ PUNCT american_scientific_journal-3517 62 33 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 34 1 1 NUM american_scientific_journal-3517 62 35 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 62 36 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 37 1 1 NUM american_scientific_journal-3517 62 38 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 62 39 βˆ’ βˆ’ PROPN american_scientific_journal-3517 62 40 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 62 41 ∞ ∞ NUM american_scientific_journal-3517 62 42 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 62 43 𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 ADJ american_scientific_journal-3517 62 44 = = SYM american_scientific_journal-3517 62 45 βˆ’ βˆ’ NOUN american_scientific_journal-3517 62 46 π‘’π‘’βˆ’π‘–π‘– π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 62 47 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 62 48 2 2 NUM american_scientific_journal-3517 62 49 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 62 50 πœ‹πœ‹ πœ‹πœ‹ PUNCT american_scientific_journal-3517 62 51 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 52 1 1 NUM american_scientific_journal-3517 62 53 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 62 54 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 55 1 1 NUM american_scientific_journal-3517 62 56 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 62 57 βˆ’ βˆ’ PROPN american_scientific_journal-3517 62 58 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 62 59 ∞ ∞ PROPN american_scientific_journal-3517 62 60 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 62 61 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 62 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 63 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 62 64 ) ) PUNCT american_scientific_journal-3517 62 65 ∞ ∞ PROPN american_scientific_journal-3517 62 66 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 62 67 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ NOUN american_scientific_journal-3517 62 68 𝑖𝑖 𝑖𝑖 ADP american_scientific_journal-3517 62 69 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 62 70 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 62 71 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 72 𝑑𝑑𝑒𝑒 𝑑𝑑𝑒𝑒 AUX american_scientific_journal-3517 62 73 ( ( PUNCT american_scientific_journal-3517 62 74 12 12 NUM american_scientific_journal-3517 62 75 ) ) PUNCT american_scientific_journal-3517 62 76 interchanging interchange VERB american_scientific_journal-3517 62 77 the the DET american_scientific_journal-3517 62 78 integration integration NOUN american_scientific_journal-3517 62 79 order order NOUN american_scientific_journal-3517 62 80 = = SYM american_scientific_journal-3517 62 81 βˆ’π‘’π‘’βˆ’π‘–π‘– βˆ’π‘’π‘’βˆ’π‘–π‘– ADJ american_scientific_journal-3517 62 82 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 ADJ american_scientific_journal-3517 62 83 2 2 NUM american_scientific_journal-3517 62 84 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 62 85 οΏ½ οΏ½ NOUN american_scientific_journal-3517 62 86 1 1 NUM american_scientific_journal-3517 62 87 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 62 88 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 89 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 62 90 ) ) PUNCT american_scientific_journal-3517 62 91 ∞ ∞ PROPN american_scientific_journal-3517 62 92 βˆ’βˆž βˆ’βˆž X american_scientific_journal-3517 62 93 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 62 94 𝑖𝑖 𝑖𝑖 NUM american_scientific_journal-3517 62 95 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 62 96 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 62 97 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 98 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 99 1 1 NUM american_scientific_journal-3517 62 100 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 62 101 βˆ’ βˆ’ PROPN american_scientific_journal-3517 62 102 𝑒𝑒 𝑒𝑒 NUM american_scientific_journal-3517 62 103 ∞ ∞ PROPN american_scientific_journal-3517 62 104 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 62 105 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘’π‘’ π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘’π‘’ PROPN american_scientific_journal-3517 62 106 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 107 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 62 108 ( ( PUNCT american_scientific_journal-3517 62 109 13 13 NUM american_scientific_journal-3517 62 110 ) ) PUNCT american_scientific_journal-3517 62 111 = = SYM american_scientific_journal-3517 62 112 βˆ’π‘’π‘’βˆ’π‘–π‘– βˆ’π‘’π‘’βˆ’π‘–π‘– X american_scientific_journal-3517 62 113 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 ADJ american_scientific_journal-3517 62 114 2 2 NUM american_scientific_journal-3517 62 115 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 62 116 οΏ½ οΏ½ NOUN american_scientific_journal-3517 62 117 1 1 NUM american_scientific_journal-3517 62 118 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– NUM american_scientific_journal-3517 62 119 οΏ½ οΏ½ PROPN american_scientific_journal-3517 62 120 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 62 121 ) ) PUNCT american_scientific_journal-3517 62 122 ∞ ∞ PROPN american_scientific_journal-3517 63 1 βˆ’βˆž βˆ’βˆž AUX american_scientific_journal-3517 63 2 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 63 3 𝑖𝑖 𝑖𝑖 X american_scientific_journal-3517 63 4 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 63 5 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠 NUM american_scientific_journal-3517 63 6 οΏ½ οΏ½ PROPN american_scientific_journal-3517 63 7 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 63 8 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 NOUN american_scientific_journal-3517 63 9 οΏ½ οΏ½ PROPN american_scientific_journal-3517 63 10 π‘’π‘’βˆ’π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– π‘’π‘’βˆ’π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– PROPN american_scientific_journal-3517 63 11 ( ( PUNCT american_scientific_journal-3517 63 12 14 14 NUM american_scientific_journal-3517 63 13 ) ) PUNCT american_scientific_journal-3517 63 14 π»π»βˆ’π‘–π‘– π»π»βˆ’π‘–π‘– NOUN american_scientific_journal-3517 63 15 οΏ½ οΏ½ NOUN american_scientific_journal-3517 63 16 𝐿𝐿𝑖𝑖 𝐿𝐿𝑖𝑖 NOUN american_scientific_journal-3517 63 17 οΏ½ οΏ½ PROPN american_scientific_journal-3517 63 18 𝐹𝐹(𝑖𝑖)𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑖𝑖2 𝐹𝐹(𝑖𝑖)𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑖𝑖2 PROPN american_scientific_journal-3517 63 19 οΏ½ οΏ½ PROPN american_scientific_journal-3517 63 20 (𝑒𝑒) (𝑒𝑒) PROPN american_scientific_journal-3517 63 21 οΏ½ οΏ½ PROPN american_scientific_journal-3517 63 22 (𝑦𝑦 (𝑦𝑦 PUNCT american_scientific_journal-3517 63 23 ) ) PUNCT american_scientific_journal-3517 63 24 = = SYM american_scientific_journal-3517 63 25 βˆ’πΏπΏπ‘–π‘– βˆ’πΏπΏπ‘–π‘– PROPN american_scientific_journal-3517 63 26 οΏ½ οΏ½ PROPN american_scientific_journal-3517 63 27 οΏ½ οΏ½ PROPN american_scientific_journal-3517 63 28 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 PROPN american_scientific_journal-3517 63 29 ) ) PUNCT american_scientific_journal-3517 63 30 ∞ ∞ PROPN american_scientific_journal-3517 64 1 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 64 2 𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠 NOUN american_scientific_journal-3517 64 3 οΏ½ οΏ½ PROPN american_scientific_journal-3517 64 4 𝑖𝑖 𝑖𝑖 SCONJ american_scientific_journal-3517 64 5 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 NOUN american_scientific_journal-3517 64 6 οΏ½ οΏ½ PROPN american_scientific_journal-3517 64 7 οΏ½ οΏ½ PROPN american_scientific_journal-3517 64 8 ( ( PUNCT american_scientific_journal-3517 64 9 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 64 10 ) ) PUNCT american_scientific_journal-3517 64 11 ( ( PUNCT american_scientific_journal-3517 64 12 15 15 NUM american_scientific_journal-3517 64 13 ) ) PUNCT american_scientific_journal-3517 64 14 equation equation NOUN american_scientific_journal-3517 64 15 ( ( PUNCT american_scientific_journal-3517 64 16 15 15 NUM american_scientific_journal-3517 64 17 ) ) PUNCT american_scientific_journal-3517 64 18 is be AUX american_scientific_journal-3517 64 19 the the DET american_scientific_journal-3517 64 20 relation relation NOUN american_scientific_journal-3517 64 21 between between ADP american_scientific_journal-3517 64 22 generalized generalized ADJ american_scientific_journal-3517 64 23 fractional fractional ADJ american_scientific_journal-3517 64 24 hilbert hilbert NOUN american_scientific_journal-3517 64 25 transform transform NOUN american_scientific_journal-3517 64 26 and and CCONJ american_scientific_journal-3517 64 27 fractional fractional ADJ american_scientific_journal-3517 64 28 laplace laplace NOUN american_scientific_journal-3517 64 29 transform transform NOUN american_scientific_journal-3517 64 30 . . PUNCT american_scientific_journal-3517 65 1 american american ADJ american_scientific_journal-3517 65 2 scientific scientific ADJ american_scientific_journal-3517 65 3 research research PROPN american_scientific_journal-3517 65 4 journal journal NOUN american_scientific_journal-3517 65 5 for for ADP american_scientific_journal-3517 65 6 engineering engineering NOUN american_scientific_journal-3517 65 7 , , PUNCT american_scientific_journal-3517 65 8 technology technology NOUN american_scientific_journal-3517 65 9 , , PUNCT american_scientific_journal-3517 65 10 and and CCONJ american_scientific_journal-3517 65 11 sciences science NOUN american_scientific_journal-3517 65 12 ( ( PUNCT american_scientific_journal-3517 65 13 asrjets asrjet NOUN american_scientific_journal-3517 65 14 ) ) PUNCT american_scientific_journal-3517 65 15 ( ( PUNCT american_scientific_journal-3517 65 16 2017 2017 NUM american_scientific_journal-3517 65 17 ) ) PUNCT american_scientific_journal-3517 65 18 volume volume NOUN american_scientific_journal-3517 65 19 38 38 NUM american_scientific_journal-3517 65 20 , , PUNCT american_scientific_journal-3517 65 21 no no DET american_scientific_journal-3517 65 22 2 2 NUM american_scientific_journal-3517 65 23 , , PUNCT american_scientific_journal-3517 65 24 pp pp ADV american_scientific_journal-3517 65 25 90 90 NUM american_scientific_journal-3517 65 26 - - SYM american_scientific_journal-3517 65 27 97 97 NUM american_scientific_journal-3517 65 28 95 95 NUM american_scientific_journal-3517 65 29 2.4 2.4 NUM american_scientific_journal-3517 65 30 the the DET american_scientific_journal-3517 65 31 relationship relationship NOUN american_scientific_journal-3517 65 32 between between ADP american_scientific_journal-3517 65 33 generalized generalized ADJ american_scientific_journal-3517 65 34 fractional fractional ADJ american_scientific_journal-3517 65 35 hilbert hilbert NOUN american_scientific_journal-3517 65 36 transform transform NOUN american_scientific_journal-3517 65 37 with with ADP american_scientific_journal-3517 65 38 fractional fractional ADJ american_scientific_journal-3517 65 39 inverse inverse NOUN american_scientific_journal-3517 65 40 laplace laplace NOUN american_scientific_journal-3517 65 41 transforms transform VERB american_scientific_journal-3517 65 42 the the DET american_scientific_journal-3517 65 43 fraction fraction NOUN american_scientific_journal-3517 65 44 inverse inverse NOUN american_scientific_journal-3517 65 45 laplace laplace NOUN american_scientific_journal-3517 65 46 transform transform NOUN american_scientific_journal-3517 65 47 is be AUX american_scientific_journal-3517 65 48 defined define VERB american_scientific_journal-3517 65 49 in in ADP american_scientific_journal-3517 65 50 [ [ X american_scientific_journal-3517 65 51 24 24 NUM american_scientific_journal-3517 65 52 ] ] SYM american_scientific_journal-3517 65 53 𝑓𝑓(𝑖𝑖 𝑓𝑓(𝑖𝑖 NOUN american_scientific_journal-3517 65 54 ) ) PUNCT american_scientific_journal-3517 65 55 = = PUNCT american_scientific_journal-3517 65 56 οΏ½ οΏ½ PROPN american_scientific_journal-3517 65 57 1 1 NUM american_scientific_journal-3517 65 58 + + CCONJ american_scientific_journal-3517 65 59 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADV american_scientific_journal-3517 65 60 2πœ‹πœ‹ 2πœ‹πœ‹ NOUN american_scientific_journal-3517 65 61 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 65 62 𝑖𝑖 𝑖𝑖 NUM american_scientific_journal-3517 65 63 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 65 64 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 65 65 οΏ½ οΏ½ NOUN american_scientific_journal-3517 65 66 𝐹𝐹 𝐹𝐹 PROPN american_scientific_journal-3517 66 1 𝑖𝑖+∞ 𝑖𝑖+∞ SCONJ american_scientific_journal-3517 66 2 π‘–π‘–βˆ’βˆž π‘–π‘–βˆ’βˆž PROPN american_scientific_journal-3517 66 3 ( ( PUNCT american_scientific_journal-3517 66 4 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ X american_scientific_journal-3517 66 5 𝑖𝑖 𝑖𝑖 X american_scientific_journal-3517 66 6 4𝑑𝑑 4𝑑𝑑 NOUN american_scientific_journal-3517 66 7 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 NUM american_scientific_journal-3517 66 8 𝐻𝐻𝑖𝑖 𝐻𝐻𝑖𝑖 ADJ american_scientific_journal-3517 66 9 οΏ½ οΏ½ NOUN american_scientific_journal-3517 66 10 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) ADJ american_scientific_journal-3517 66 11 οΏ½ οΏ½ PROPN american_scientific_journal-3517 66 12 (𝑦𝑦 (𝑦𝑦 PUNCT american_scientific_journal-3517 66 13 ) ) PUNCT american_scientific_journal-3517 66 14 = = SYM american_scientific_journal-3517 66 15 π‘’π‘’βˆ’ π‘’π‘’βˆ’ SYM american_scientific_journal-3517 66 16 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 VERB american_scientific_journal-3517 66 17 2 2 NUM american_scientific_journal-3517 66 18 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 66 19 πœ‹πœ‹ πœ‹πœ‹ X american_scientific_journal-3517 66 20 οΏ½ οΏ½ PROPN american_scientific_journal-3517 66 21 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 ADV american_scientific_journal-3517 66 22 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖 PROPN american_scientific_journal-3517 66 23 ) ) PUNCT american_scientific_journal-3517 66 24 𝑦𝑦 𝑦𝑦 ADP american_scientific_journal-3517 66 25 βˆ’ βˆ’ PROPN american_scientific_journal-3517 66 26 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 66 27 ∞ ∞ PROPN american_scientific_journal-3517 66 28 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 66 29 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 66 30 𝑖𝑖 𝑖𝑖 X american_scientific_journal-3517 66 31 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 66 32 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 PROPN american_scientific_journal-3517 66 33 = = SYM american_scientific_journal-3517 66 34 π‘’π‘’βˆ’ π‘’π‘’βˆ’ SYM american_scientific_journal-3517 66 35 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 VERB american_scientific_journal-3517 66 36 2 2 NUM american_scientific_journal-3517 66 37 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 66 38 πœ‹πœ‹ πœ‹πœ‹ X american_scientific_journal-3517 66 39 οΏ½ οΏ½ PROPN american_scientific_journal-3517 66 40 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 VERB american_scientific_journal-3517 66 41 𝑦𝑦 𝑦𝑦 ADP american_scientific_journal-3517 66 42 βˆ’ βˆ’ PROPN american_scientific_journal-3517 66 43 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 66 44 ∞ ∞ PROPN american_scientific_journal-3517 66 45 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 66 46 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 66 47 𝑖𝑖 𝑖𝑖 NUM american_scientific_journal-3517 66 48 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 66 49 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 66 50 οΏ½ οΏ½ PROPN american_scientific_journal-3517 66 51 οΏ½ οΏ½ PROPN american_scientific_journal-3517 66 52 1 1 NUM american_scientific_journal-3517 66 53 + + CCONJ american_scientific_journal-3517 66 54 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADV american_scientific_journal-3517 66 55 2πœ‹πœ‹ 2πœ‹πœ‹ NOUN american_scientific_journal-3517 66 56 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 66 57 𝑖𝑖 𝑖𝑖 NUM american_scientific_journal-3517 66 58 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 66 59 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 66 60 οΏ½ οΏ½ PROPN american_scientific_journal-3517 66 61 𝐹𝐹𝐿𝐿.𝑖𝑖 𝐹𝐹𝐿𝐿.𝑖𝑖 PROPN american_scientific_journal-3517 66 62 𝑖𝑖+∞ 𝑖𝑖+∞ PRON american_scientific_journal-3517 66 63 π‘–π‘–βˆ’βˆž π‘–π‘–βˆ’βˆž PROPN american_scientific_journal-3517 66 64 ( ( PUNCT american_scientific_journal-3517 66 65 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ NOUN american_scientific_journal-3517 66 66 𝑖𝑖𝑖𝑖2 𝑖𝑖𝑖𝑖2 NOUN american_scientific_journal-3517 66 67 4 4 NUM american_scientific_journal-3517 67 1 𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 NOUN american_scientific_journal-3517 67 2 οΏ½ οΏ½ NOUN american_scientific_journal-3517 67 3 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 67 4 = = SYM american_scientific_journal-3517 67 5 π‘’π‘’βˆ’ π‘’π‘’βˆ’ PUNCT american_scientific_journal-3517 67 6 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 X american_scientific_journal-3517 67 7 2 2 NUM american_scientific_journal-3517 67 8 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 67 9 οΏ½ οΏ½ NOUN american_scientific_journal-3517 67 10 1 1 NUM american_scientific_journal-3517 67 11 + + CCONJ american_scientific_journal-3517 67 12 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 67 13 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 67 14 οΏ½ οΏ½ PROPN american_scientific_journal-3517 67 15 𝐹𝐹𝐿𝐿.𝑖𝑖 𝐹𝐹𝐿𝐿.𝑖𝑖 PROPN american_scientific_journal-3517 67 16 𝑖𝑖+∞ 𝑖𝑖+∞ PRON american_scientific_journal-3517 67 17 π‘–π‘–βˆ’βˆž π‘–π‘–βˆ’βˆž PROPN american_scientific_journal-3517 67 18 ( ( PUNCT american_scientific_journal-3517 67 19 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ PROPN american_scientific_journal-3517 67 20 𝑖𝑖 𝑖𝑖 PUNCT american_scientific_journal-3517 67 21 4𝑑𝑑 4𝑑𝑑 ADJ american_scientific_journal-3517 67 22 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 NUM american_scientific_journal-3517 67 23 οΏ½ οΏ½ PROPN american_scientific_journal-3517 67 24 1 1 NUM american_scientific_journal-3517 67 25 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 67 26 οΏ½ οΏ½ PROPN american_scientific_journal-3517 67 27 𝑒𝑒𝑖𝑖𝑑𝑑 𝑒𝑒𝑖𝑖𝑑𝑑 PROPN american_scientific_journal-3517 67 28 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 67 29 βˆ’ βˆ’ PROPN american_scientific_journal-3517 67 30 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 67 31 ∞ ∞ PROPN american_scientific_journal-3517 67 32 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 67 33 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 ADJ american_scientific_journal-3517 67 34 οΏ½ οΏ½ NOUN american_scientific_journal-3517 67 35 𝑑𝑑𝑒𝑒 𝑑𝑑𝑒𝑒 NOUN american_scientific_journal-3517 67 36 = = SYM american_scientific_journal-3517 67 37 π‘’π‘’βˆ’ π‘’π‘’βˆ’ X american_scientific_journal-3517 67 38 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 VERB american_scientific_journal-3517 67 39 2 2 NUM american_scientific_journal-3517 67 40 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 67 41 οΏ½ οΏ½ NOUN american_scientific_journal-3517 67 42 1 1 NUM american_scientific_journal-3517 67 43 + + CCONJ american_scientific_journal-3517 67 44 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 67 45 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 67 46 οΏ½ οΏ½ PROPN american_scientific_journal-3517 67 47 𝐹𝐹𝐿𝐿.𝑖𝑖 𝐹𝐹𝐿𝐿.𝑖𝑖 PROPN american_scientific_journal-3517 67 48 𝑖𝑖+∞ 𝑖𝑖+∞ PRON american_scientific_journal-3517 67 49 π‘–π‘–βˆ’βˆž π‘–π‘–βˆ’βˆž PROPN american_scientific_journal-3517 67 50 ( ( PUNCT american_scientific_journal-3517 67 51 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ PROPN american_scientific_journal-3517 67 52 𝑖𝑖 𝑖𝑖 PUNCT american_scientific_journal-3517 67 53 4𝑑𝑑 4𝑑𝑑 NOUN american_scientific_journal-3517 67 54 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 NUM american_scientific_journal-3517 67 55 . . PUNCT american_scientific_journal-3517 68 1 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 VERB american_scientific_journal-3517 68 2 𝐻𝐻𝑖𝑖 𝐻𝐻𝑖𝑖 NOUN american_scientific_journal-3517 68 3 οΏ½ οΏ½ NOUN american_scientific_journal-3517 68 4 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) ADJ american_scientific_journal-3517 68 5 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 6 (𝑦𝑦 (𝑦𝑦 PUNCT american_scientific_journal-3517 68 7 ) ) PUNCT american_scientific_journal-3517 68 8 = = SYM american_scientific_journal-3517 68 9 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑦𝑦 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑦𝑦 PROPN american_scientific_journal-3517 68 10 ) ) PUNCT american_scientific_journal-3517 68 11 on on ADP american_scientific_journal-3517 68 12 the the DET american_scientific_journal-3517 68 13 relationship relationship NOUN american_scientific_journal-3517 68 14 between between ADP american_scientific_journal-3517 68 15 generalized generalized ADJ american_scientific_journal-3517 68 16 inverse inverse NOUN american_scientific_journal-3517 68 17 fractional fractional ADJ american_scientific_journal-3517 68 18 hilbert hilbert NOUN american_scientific_journal-3517 68 19 transform transform NOUN american_scientific_journal-3517 68 20 with with ADP american_scientific_journal-3517 68 21 fractional fractional ADJ american_scientific_journal-3517 68 22 inverse inverse NOUN american_scientific_journal-3517 68 23 laplace laplace NOUN american_scientific_journal-3517 68 24 transform transform NOUN american_scientific_journal-3517 68 25 π»π»βˆ’π‘–π‘– π»π»βˆ’π‘–π‘– PROPN american_scientific_journal-3517 68 26 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 27 𝑒𝑒𝑖𝑖𝑑𝑑2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) 𝑒𝑒𝑖𝑖𝑑𝑑2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) NOUN american_scientific_journal-3517 68 28 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 29 (𝑦𝑦 (𝑦𝑦 PUNCT american_scientific_journal-3517 68 30 ) ) PUNCT american_scientific_journal-3517 68 31 = = SYM american_scientific_journal-3517 68 32 βˆ’π‘’π‘’ βˆ’π‘’π‘’ X american_scientific_journal-3517 68 33 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 X american_scientific_journal-3517 68 34 2 2 NUM american_scientific_journal-3517 68 35 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 68 36 πœ‹πœ‹ πœ‹πœ‹ X american_scientific_journal-3517 68 37 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 38 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 ADV american_scientific_journal-3517 68 39 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖 PROPN american_scientific_journal-3517 68 40 ) ) PUNCT american_scientific_journal-3517 68 41 𝑦𝑦 𝑦𝑦 ADP american_scientific_journal-3517 68 42 βˆ’ βˆ’ PROPN american_scientific_journal-3517 68 43 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 68 44 ∞ ∞ PROPN american_scientific_journal-3517 68 45 βˆ’βˆž βˆ’βˆž X american_scientific_journal-3517 68 46 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 68 47 βˆ’π‘–π‘– βˆ’π‘–π‘– PROPN american_scientific_journal-3517 68 48 2 2 NUM american_scientific_journal-3517 68 49 𝑑𝑑 𝑑𝑑 PROPN american_scientific_journal-3517 68 50 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 68 51 = = SYM american_scientific_journal-3517 68 52 βˆ’π‘’π‘’ βˆ’π‘’π‘’ X american_scientific_journal-3517 68 53 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 X american_scientific_journal-3517 68 54 2 2 NUM american_scientific_journal-3517 68 55 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 68 56 πœ‹πœ‹ πœ‹πœ‹ X american_scientific_journal-3517 68 57 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 58 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 VERB american_scientific_journal-3517 68 59 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 68 60 βˆ’ βˆ’ PROPN american_scientific_journal-3517 68 61 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 68 62 ∞ ∞ PROPN american_scientific_journal-3517 68 63 βˆ’βˆž βˆ’βˆž X american_scientific_journal-3517 68 64 π‘’π‘’βˆ’ π‘’π‘’βˆ’ X american_scientific_journal-3517 68 65 𝑖𝑖 𝑖𝑖 NUM american_scientific_journal-3517 68 66 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 68 67 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 68 68 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 69 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 70 1 1 NUM american_scientific_journal-3517 68 71 + + CCONJ american_scientific_journal-3517 68 72 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADV american_scientific_journal-3517 68 73 2πœ‹πœ‹ 2πœ‹πœ‹ NOUN american_scientific_journal-3517 68 74 𝑒𝑒 𝑒𝑒 ADP american_scientific_journal-3517 68 75 𝑖𝑖 𝑖𝑖 NUM american_scientific_journal-3517 68 76 2𝑑𝑑 2𝑑𝑑 PROPN american_scientific_journal-3517 68 77 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 NUM american_scientific_journal-3517 68 78 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 79 𝐹𝐹𝐿𝐿.𝑖𝑖 𝐹𝐹𝐿𝐿.𝑖𝑖 PROPN american_scientific_journal-3517 68 80 𝑖𝑖+∞ 𝑖𝑖+∞ PRON american_scientific_journal-3517 68 81 π‘–π‘–βˆ’βˆž π‘–π‘–βˆ’βˆž PROPN american_scientific_journal-3517 68 82 ( ( PUNCT american_scientific_journal-3517 68 83 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ NOUN american_scientific_journal-3517 68 84 𝑖𝑖𝑖𝑖2 𝑖𝑖𝑖𝑖2 NOUN american_scientific_journal-3517 68 85 4 4 NUM american_scientific_journal-3517 68 86 𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 NOUN american_scientific_journal-3517 68 87 οΏ½ οΏ½ NOUN american_scientific_journal-3517 68 88 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 NOUN american_scientific_journal-3517 68 89 = = SYM american_scientific_journal-3517 68 90 βˆ’π‘’π‘’ βˆ’π‘’π‘’ PROPN american_scientific_journal-3517 68 91 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 X american_scientific_journal-3517 68 92 2 2 NUM american_scientific_journal-3517 68 93 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 68 94 οΏ½ οΏ½ NOUN american_scientific_journal-3517 68 95 1 1 NUM american_scientific_journal-3517 68 96 + + CCONJ american_scientific_journal-3517 68 97 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 68 98 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 68 99 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 100 𝐹𝐹𝐿𝐿.𝑖𝑖 𝐹𝐹𝐿𝐿.𝑖𝑖 PROPN american_scientific_journal-3517 68 101 𝑖𝑖+∞ 𝑖𝑖+∞ PRON american_scientific_journal-3517 68 102 π‘–π‘–βˆ’βˆž π‘–π‘–βˆ’βˆž PROPN american_scientific_journal-3517 68 103 ( ( PUNCT american_scientific_journal-3517 68 104 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ PROPN american_scientific_journal-3517 68 105 𝑖𝑖 𝑖𝑖 PUNCT american_scientific_journal-3517 68 106 4𝑑𝑑 4𝑑𝑑 ADJ american_scientific_journal-3517 68 107 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 NUM american_scientific_journal-3517 68 108 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 109 1 1 NUM american_scientific_journal-3517 68 110 πœ‹πœ‹ πœ‹πœ‹ PROPN american_scientific_journal-3517 68 111 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 112 𝑒𝑒𝑖𝑖𝑑𝑑 𝑒𝑒𝑖𝑖𝑑𝑑 PROPN american_scientific_journal-3517 68 113 𝑦𝑦 𝑦𝑦 PROPN american_scientific_journal-3517 68 114 βˆ’ βˆ’ PROPN american_scientific_journal-3517 68 115 𝑖𝑖 𝑖𝑖 NOUN american_scientific_journal-3517 68 116 ∞ ∞ PROPN american_scientific_journal-3517 68 117 βˆ’βˆž βˆ’βˆž ADP american_scientific_journal-3517 68 118 𝑑𝑑𝑖𝑖 𝑑𝑑𝑖𝑖 ADJ american_scientific_journal-3517 68 119 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 120 𝑑𝑑𝑒𝑒 𝑑𝑑𝑒𝑒 NOUN american_scientific_journal-3517 68 121 = = SYM american_scientific_journal-3517 68 122 βˆ’π‘’π‘’ βˆ’π‘’π‘’ X american_scientific_journal-3517 68 123 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 X american_scientific_journal-3517 68 124 2 2 NUM american_scientific_journal-3517 68 125 𝑦𝑦2 𝑦𝑦2 NOUN american_scientific_journal-3517 68 126 οΏ½ οΏ½ NOUN american_scientific_journal-3517 68 127 1 1 NUM american_scientific_journal-3517 68 128 + + CCONJ american_scientific_journal-3517 68 129 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ADJ american_scientific_journal-3517 68 130 2πœ‹πœ‹ 2πœ‹πœ‹ ADJ american_scientific_journal-3517 68 131 οΏ½ οΏ½ PROPN american_scientific_journal-3517 68 132 𝐹𝐹𝐿𝐿.𝑖𝑖 𝐹𝐹𝐿𝐿.𝑖𝑖 PROPN american_scientific_journal-3517 68 133 𝑖𝑖+∞ 𝑖𝑖+∞ PRON american_scientific_journal-3517 68 134 π‘–π‘–βˆ’βˆž π‘–π‘–βˆ’βˆž PROPN american_scientific_journal-3517 68 135 ( ( PUNCT american_scientific_journal-3517 68 136 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ PROPN american_scientific_journal-3517 68 137 𝑖𝑖 𝑖𝑖 PUNCT american_scientific_journal-3517 68 138 4𝑑𝑑 4𝑑𝑑 NOUN american_scientific_journal-3517 68 139 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 NUM american_scientific_journal-3517 68 140 . . PUNCT american_scientific_journal-3517 69 1 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 VERB american_scientific_journal-3517 69 2 𝐻𝐻𝑖𝑖 𝐻𝐻𝑖𝑖 NOUN american_scientific_journal-3517 69 3 οΏ½ οΏ½ NOUN american_scientific_journal-3517 69 4 π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) ADJ american_scientific_journal-3517 69 5 οΏ½ οΏ½ PROPN american_scientific_journal-3517 69 6 (𝑦𝑦 (𝑦𝑦 PUNCT american_scientific_journal-3517 69 7 ) ) PUNCT american_scientific_journal-3517 69 8 = = SYM american_scientific_journal-3517 69 9 βˆ’πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑦𝑦 βˆ’πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑦𝑦 NOUN american_scientific_journal-3517 69 10 ) ) PUNCT american_scientific_journal-3517 69 11 american american ADJ american_scientific_journal-3517 69 12 scientific scientific ADJ american_scientific_journal-3517 69 13 research research NOUN american_scientific_journal-3517 69 14 journal journal NOUN american_scientific_journal-3517 69 15 for for ADP american_scientific_journal-3517 69 16 engineering engineering NOUN american_scientific_journal-3517 69 17 , , PUNCT american_scientific_journal-3517 69 18 technology technology NOUN american_scientific_journal-3517 69 19 , , PUNCT american_scientific_journal-3517 69 20 and and CCONJ american_scientific_journal-3517 69 21 sciences science NOUN american_scientific_journal-3517 69 22 ( ( PUNCT american_scientific_journal-3517 69 23 asrjets asrjet NOUN american_scientific_journal-3517 69 24 ) ) PUNCT american_scientific_journal-3517 69 25 ( ( PUNCT american_scientific_journal-3517 69 26 2017 2017 NUM american_scientific_journal-3517 69 27 ) ) PUNCT american_scientific_journal-3517 69 28 volume volume NOUN american_scientific_journal-3517 69 29 38 38 NUM american_scientific_journal-3517 69 30 , , PUNCT american_scientific_journal-3517 69 31 no no DET american_scientific_journal-3517 69 32 2 2 NUM american_scientific_journal-3517 69 33 , , PUNCT american_scientific_journal-3517 69 34 pp pp ADV american_scientific_journal-3517 69 35 90 90 NUM american_scientific_journal-3517 69 36 - - SYM american_scientific_journal-3517 69 37 97 97 NUM american_scientific_journal-3517 69 38 96 96 NUM american_scientific_journal-3517 69 39 3 3 NUM american_scientific_journal-3517 69 40 . . PUNCT american_scientific_journal-3517 70 1 conclusion conclusion NOUN american_scientific_journal-3517 70 2 we we PRON american_scientific_journal-3517 70 3 have have AUX american_scientific_journal-3517 70 4 established establish VERB american_scientific_journal-3517 70 5 mathematically mathematically ADV american_scientific_journal-3517 70 6 the the DET american_scientific_journal-3517 70 7 relationship relationship NOUN american_scientific_journal-3517 70 8 of of ADP american_scientific_journal-3517 70 9 the the DET american_scientific_journal-3517 70 10 fractional fractional ADJ american_scientific_journal-3517 70 11 hilbert hilbert NOUN american_scientific_journal-3517 70 12 transform transform NOUN american_scientific_journal-3517 70 13 with with ADP american_scientific_journal-3517 70 14 fractional fractional ADJ american_scientific_journal-3517 70 15 mellin mellin PROPN american_scientific_journal-3517 70 16 transform transform NOUN american_scientific_journal-3517 70 17 and and CCONJ american_scientific_journal-3517 70 18 fractional fractional ADJ american_scientific_journal-3517 70 19 laplace laplace NOUN american_scientific_journal-3517 70 20 transform transform NOUN american_scientific_journal-3517 70 21 . . PUNCT american_scientific_journal-3517 70 22 , , PUNCT american_scientific_journal-3517 70 23 which which PRON american_scientific_journal-3517 70 24 will will AUX american_scientific_journal-3517 70 25 play play VERB american_scientific_journal-3517 70 26 a a DET american_scientific_journal-3517 70 27 significant significant ADJ american_scientific_journal-3517 70 28 role role NOUN american_scientific_journal-3517 70 29 in in ADP american_scientific_journal-3517 70 30 signal signal ADJ american_scientific_journal-3517 70 31 processing processing NOUN american_scientific_journal-3517 70 32 and and CCONJ american_scientific_journal-3517 70 33 other other ADJ american_scientific_journal-3517 70 34 field field NOUN american_scientific_journal-3517 70 35 of of ADP american_scientific_journal-3517 70 36 applied applied ADJ american_scientific_journal-3517 70 37 mathematics mathematic NOUN american_scientific_journal-3517 70 38 , , PUNCT american_scientific_journal-3517 70 39 engineering engineering NOUN american_scientific_journal-3517 70 40 and and CCONJ american_scientific_journal-3517 70 41 physics physics NOUN american_scientific_journal-3517 70 42 . . PUNCT american_scientific_journal-3517 71 1 references reference NOUN american_scientific_journal-3517 71 2 [ [ X american_scientific_journal-3517 71 3 1 1 NUM american_scientific_journal-3517 71 4 ] ] X american_scientific_journal-3517 71 5 a.w a.w PROPN american_scientific_journal-3517 71 6 . . PROPN american_scientific_journal-3517 71 7 lohmann lohmann PROPN american_scientific_journal-3517 71 8 , , PUNCT american_scientific_journal-3517 71 9 d. d. PROPN american_scientific_journal-3517 71 10 mendlovic mendlovic PROPN american_scientific_journal-3517 71 11 , , PUNCT american_scientific_journal-3517 71 12 z. z. PROPN american_scientific_journal-3517 71 13 zalevsky zalevsky PROPN american_scientific_journal-3517 71 14 , , PUNCT american_scientific_journal-3517 71 15 fractional fractional ADJ american_scientific_journal-3517 71 16 hilbert hilbert NOUN american_scientific_journal-3517 71 17 transform transform NOUN american_scientific_journal-3517 71 18 , , PUNCT american_scientific_journal-3517 71 19 optics optic NOUN american_scientific_journal-3517 71 20 letters letter NOUN american_scientific_journal-3517 71 21 21 21 NUM american_scientific_journal-3517 71 22 ( ( PUNCT american_scientific_journal-3517 71 23 1996 1996 NUM american_scientific_journal-3517 71 24 ) ) PUNCT american_scientific_journal-3517 71 25 281–283 281–283 NUM american_scientific_journal-3517 71 26 . . PUNCT american_scientific_journal-3517 72 1 [ [ X american_scientific_journal-3517 72 2 2 2 X american_scientific_journal-3517 72 3 ] ] X american_scientific_journal-3517 72 4 j.a j.a PROPN american_scientific_journal-3517 72 5 . . PROPN american_scientific_journal-3517 72 6 davis davis PROPN american_scientific_journal-3517 72 7 , , PUNCT american_scientific_journal-3517 72 8 d.e d.e PROPN american_scientific_journal-3517 72 9 . . PROPN american_scientific_journal-3517 72 10 mcnamara mcnamara PROPN american_scientific_journal-3517 72 11 , , PUNCT american_scientific_journal-3517 72 12 d.m d.m PROPN american_scientific_journal-3517 72 13 . . PUNCT american_scientific_journal-3517 72 14 cottrell cottrell PROPN american_scientific_journal-3517 72 15 , , PUNCT american_scientific_journal-3517 72 16 analysis analysis NOUN american_scientific_journal-3517 72 17 of of ADP american_scientific_journal-3517 72 18 the the DET american_scientific_journal-3517 72 19 fractional fractional ADJ american_scientific_journal-3517 72 20 hilbert hilbert NOUN american_scientific_journal-3517 72 21 transform transform NOUN american_scientific_journal-3517 72 22 , , PUNCT american_scientific_journal-3517 72 23 applied apply VERB american_scientific_journal-3517 72 24 optics optic NOUN american_scientific_journal-3517 72 25 37 37 NUM american_scientific_journal-3517 72 26 ( ( PUNCT american_scientific_journal-3517 72 27 1998 1998 NUM american_scientific_journal-3517 72 28 ) ) PUNCT american_scientific_journal-3517 72 29 6911 6911 NUM american_scientific_journal-3517 72 30 - - SYM american_scientific_journal-3517 72 31 6913 6913 NUM american_scientific_journal-3517 73 1 [ [ X american_scientific_journal-3517 73 2 3 3 NUM american_scientific_journal-3517 73 3 ] ] PUNCT american_scientific_journal-3517 73 4 l. l. PROPN american_scientific_journal-3517 73 5 cohen cohen PROPN american_scientific_journal-3517 73 6 , , PUNCT american_scientific_journal-3517 73 7 p. p. PROPN american_scientific_journal-3517 73 8 loughlin loughlin PROPN american_scientific_journal-3517 73 9 , , PUNCT american_scientific_journal-3517 73 10 d. d. PROPN american_scientific_journal-3517 73 11 vakman vakman PROPN american_scientific_journal-3517 73 12 , , PUNCT american_scientific_journal-3517 73 13 on on ADP american_scientific_journal-3517 73 14 an an DET american_scientific_journal-3517 73 15 ambiguity ambiguity NOUN american_scientific_journal-3517 73 16 in in ADP american_scientific_journal-3517 73 17 the the DET american_scientific_journal-3517 73 18 definition definition NOUN american_scientific_journal-3517 73 19 of of ADP american_scientific_journal-3517 73 20 the the DET american_scientific_journal-3517 73 21 amplitude amplitude NOUN american_scientific_journal-3517 73 22 and and CCONJ american_scientific_journal-3517 73 23 phase phase NOUN american_scientific_journal-3517 73 24 of of ADP american_scientific_journal-3517 73 25 a a DET american_scientific_journal-3517 73 26 signal signal NOUN american_scientific_journal-3517 73 27 , , PUNCT american_scientific_journal-3517 73 28 signal signal NOUN american_scientific_journal-3517 73 29 processing process VERB american_scientific_journal-3517 73 30 79 79 NUM american_scientific_journal-3517 73 31 ( ( PUNCT american_scientific_journal-3517 73 32 1999 1999 NUM american_scientific_journal-3517 73 33 ) ) PUNCT american_scientific_journal-3517 73 34 301–307 301–307 NUM american_scientific_journal-3517 73 35 . . PUNCT american_scientific_journal-3517 74 1 [ [ X american_scientific_journal-3517 74 2 4 4 NUM american_scientific_journal-3517 74 3 ] ] X american_scientific_journal-3517 74 4 a.i a.i PROPN american_scientific_journal-3517 74 5 . . PROPN american_scientific_journal-3517 74 6 zayed zayed PROPN american_scientific_journal-3517 74 7 , , PUNCT american_scientific_journal-3517 74 8 a.i a.i PROPN american_scientific_journal-3517 74 9 . . PROPN american_scientific_journal-3517 74 10 zayed zayed PROPN american_scientific_journal-3517 74 11 , , PUNCT american_scientific_journal-3517 74 12 hilbert hilbert NOUN american_scientific_journal-3517 74 13 transform transform NOUN american_scientific_journal-3517 74 14 associated associate VERB american_scientific_journal-3517 74 15 with with ADP american_scientific_journal-3517 74 16 the the DET american_scientific_journal-3517 74 17 fractional fractional ADJ american_scientific_journal-3517 74 18 fourier fourier NOUN american_scientific_journal-3517 74 19 transform transform NOUN american_scientific_journal-3517 74 20 , , PUNCT american_scientific_journal-3517 74 21 ieee ieee NOUN american_scientific_journal-3517 74 22 signal signal NOUN american_scientific_journal-3517 74 23 processing processing NOUN american_scientific_journal-3517 74 24 letters letter NOUN american_scientific_journal-3517 74 25 5 5 NUM american_scientific_journal-3517 74 26 ( ( PUNCT american_scientific_journal-3517 74 27 1998 1998 NUM american_scientific_journal-3517 74 28 ) ) PUNCT american_scientific_journal-3517 74 29 206 206 NUM american_scientific_journal-3517 74 30 - - SYM american_scientific_journal-3517 74 31 208 208 NUM american_scientific_journal-3517 74 32 . . PUNCT american_scientific_journal-3517 75 1 [ [ X american_scientific_journal-3517 75 2 5 5 NUM american_scientific_journal-3517 75 3 ] ] X american_scientific_journal-3517 75 4 c.c c.c PROPN american_scientific_journal-3517 75 5 . . PROPN american_scientific_journal-3517 75 6 tseng tseng PROPN american_scientific_journal-3517 75 7 , , PUNCT american_scientific_journal-3517 75 8 s.c s.c PROPN american_scientific_journal-3517 75 9 . . PROPN american_scientific_journal-3517 75 10 pei pei PROPN american_scientific_journal-3517 75 11 , , PUNCT american_scientific_journal-3517 75 12 design design NOUN american_scientific_journal-3517 75 13 and and CCONJ american_scientific_journal-3517 75 14 application application NOUN american_scientific_journal-3517 75 15 of of ADP american_scientific_journal-3517 75 16 discrete discrete ADJ american_scientific_journal-3517 75 17 - - PUNCT american_scientific_journal-3517 75 18 time time NOUN american_scientific_journal-3517 75 19 fractional fractional ADJ american_scientific_journal-3517 75 20 hilbert hilbert PROPN american_scientific_journal-3517 75 21 transformer transformer NOUN american_scientific_journal-3517 75 22 , , PUNCT american_scientific_journal-3517 75 23 ieee ieee NOUN american_scientific_journal-3517 75 24 transactions transaction NOUN american_scientific_journal-3517 75 25 on on ADP american_scientific_journal-3517 75 26 circuits circuit NOUN american_scientific_journal-3517 75 27 and and CCONJ american_scientific_journal-3517 75 28 systems system NOUN american_scientific_journal-3517 75 29 ii ii PROPN american_scientific_journal-3517 75 30 : : PUNCT american_scientific_journal-3517 75 31 analog analog NOUN american_scientific_journal-3517 75 32 and and CCONJ american_scientific_journal-3517 75 33 digital digital ADJ american_scientific_journal-3517 75 34 signal signal NOUN american_scientific_journal-3517 75 35 processing process VERB american_scientific_journal-3517 75 36 47 47 NUM american_scientific_journal-3517 75 37 ( ( PUNCT american_scientific_journal-3517 75 38 2000 2000 NUM american_scientific_journal-3517 75 39 ) ) PUNCT american_scientific_journal-3517 75 40 1529–1533 1529–1533 NUM american_scientific_journal-3517 75 41 . . PUNCT american_scientific_journal-3517 76 1 [ [ X american_scientific_journal-3517 76 2 6 6 NUM american_scientific_journal-3517 76 3 ] ] X american_scientific_journal-3517 76 4 s.c s.c PROPN american_scientific_journal-3517 76 5 . . PROPN american_scientific_journal-3517 76 6 pei pei PROPN american_scientific_journal-3517 76 7 , , PUNCT american_scientific_journal-3517 76 8 p.h p.h PROPN american_scientific_journal-3517 76 9 . . PROPN american_scientific_journal-3517 76 10 wang wang PROPN american_scientific_journal-3517 76 11 , , PUNCT american_scientific_journal-3517 76 12 analytical analytical ADJ american_scientific_journal-3517 76 13 design design NOUN american_scientific_journal-3517 76 14 of of ADP american_scientific_journal-3517 76 15 maximally maximally ADV american_scientific_journal-3517 76 16 flat flat ADJ american_scientific_journal-3517 76 17 fir fir NOUN american_scientific_journal-3517 76 18 fractional fractional ADJ american_scientific_journal-3517 76 19 hilbert hilbert NOUN american_scientific_journal-3517 76 20 transformers transformer NOUN american_scientific_journal-3517 76 21 , , PUNCT american_scientific_journal-3517 76 22 signal signal ADJ american_scientific_journal-3517 76 23 processing processing NOUN american_scientific_journal-3517 76 24 81 81 NUM american_scientific_journal-3517 76 25 ( ( PUNCT american_scientific_journal-3517 76 26 2001 2001 NUM american_scientific_journal-3517 76 27 ) ) PUNCT american_scientific_journal-3517 76 28 643–4661 643–4661 NUM american_scientific_journal-3517 76 29 . . PUNCT american_scientific_journal-3517 77 1 [ [ X american_scientific_journal-3517 77 2 7 7 NUM american_scientific_journal-3517 77 3 ] ] PUNCT american_scientific_journal-3517 77 4 a. a. NOUN american_scientific_journal-3517 77 5 cusmariu cusmariu PROPN american_scientific_journal-3517 77 6 , , PUNCT american_scientific_journal-3517 77 7 fractional fractional ADJ american_scientific_journal-3517 77 8 analytic analytic ADJ american_scientific_journal-3517 77 9 signals signal NOUN american_scientific_journal-3517 77 10 , , PUNCT american_scientific_journal-3517 77 11 signal signal ADJ american_scientific_journal-3517 77 12 processing processing NOUN american_scientific_journal-3517 77 13 82 82 NUM american_scientific_journal-3517 77 14 ( ( PUNCT american_scientific_journal-3517 77 15 2002 2002 NUM american_scientific_journal-3517 77 16 ) ) PUNCT american_scientific_journal-3517 77 17 267–272 267–272 NUM american_scientific_journal-3517 77 18 . . PUNCT american_scientific_journal-3517 78 1 [ [ X american_scientific_journal-3517 78 2 8 8 NUM american_scientific_journal-3517 78 3 ] ] X american_scientific_journal-3517 78 4 r. r. PROPN american_scientific_journal-3517 78 5 tao tao PROPN american_scientific_journal-3517 78 6 , , PUNCT american_scientific_journal-3517 78 7 x.m x.m PROPN american_scientific_journal-3517 78 8 . . PROPN american_scientific_journal-3517 78 9 li li PROPN american_scientific_journal-3517 78 10 , , PUNCT american_scientific_journal-3517 78 11 y. y. PROPN american_scientific_journal-3517 78 12 wang wang PROPN american_scientific_journal-3517 78 13 , , PUNCT american_scientific_journal-3517 78 14 generalization generalization NOUN american_scientific_journal-3517 78 15 of of ADP american_scientific_journal-3517 78 16 the the DET american_scientific_journal-3517 78 17 fractional fractional ADJ american_scientific_journal-3517 78 18 hilbert hilbert PROPN american_scientific_journal-3517 78 19 transform transform NOUN american_scientific_journal-3517 78 20 , , PUNCT american_scientific_journal-3517 78 21 ieee ieee NOUN american_scientific_journal-3517 78 22 signal signal PROPN american_scientific_journal-3517 78 23 processingletters processingletter NOUN american_scientific_journal-3517 78 24 15 15 NUM american_scientific_journal-3517 78 25 ( ( PUNCT american_scientific_journal-3517 78 26 2008 2008 NUM american_scientific_journal-3517 78 27 ) ) PUNCT american_scientific_journal-3517 79 1 365–368 365–368 NUM american_scientific_journal-3517 79 2 . . PUNCT american_scientific_journal-3517 80 1 [ [ X american_scientific_journal-3517 80 2 9 9 NUM american_scientific_journal-3517 80 3 ] ] X american_scientific_journal-3517 80 4 y. y. PROPN american_scientific_journal-3517 80 5 fu fu PROPN american_scientific_journal-3517 80 6 , , PUNCT american_scientific_journal-3517 80 7 l. l. PROPN american_scientific_journal-3517 80 8 li li PROPN american_scientific_journal-3517 80 9 , , PUNCT american_scientific_journal-3517 80 10 generalized generalized ADJ american_scientific_journal-3517 80 11 analytic analytic ADJ american_scientific_journal-3517 80 12 signal signal NOUN american_scientific_journal-3517 80 13 associated associate VERB american_scientific_journal-3517 80 14 with with ADP american_scientific_journal-3517 80 15 linear linear ADJ american_scientific_journal-3517 80 16 canonical canonical ADJ american_scientific_journal-3517 80 17 transform transform NOUN american_scientific_journal-3517 80 18 , , PUNCT american_scientific_journal-3517 80 19 optics optic NOUN american_scientific_journal-3517 80 20 communications communication NOUN american_scientific_journal-3517 80 21 281 281 NUM american_scientific_journal-3517 80 22 ( ( PUNCT american_scientific_journal-3517 80 23 2008 2008 NUM american_scientific_journal-3517 80 24 ) ) PUNCT american_scientific_journal-3517 80 25 1468–1472 1468–1472 NUM american_scientific_journal-3517 80 26 . . PUNCT american_scientific_journal-3517 81 1 [ [ X american_scientific_journal-3517 81 2 10 10 NUM american_scientific_journal-3517 81 3 ] ] PUNCT american_scientific_journal-3517 81 4 x. x. NOUN american_scientific_journal-3517 81 5 guanlei guanlei PROPN american_scientific_journal-3517 81 6 , , PUNCT american_scientific_journal-3517 81 7 w. w. PROPN american_scientific_journal-3517 81 8 xiaotong xiaotong PROPN american_scientific_journal-3517 81 9 , , PUNCT american_scientific_journal-3517 81 10 x. x. PROPN american_scientific_journal-3517 81 11 xiaogang xiaogang PROPN american_scientific_journal-3517 81 12 , , PUNCT american_scientific_journal-3517 81 13 generalized generalized ADJ american_scientific_journal-3517 81 14 hilbert hilbert NOUN american_scientific_journal-3517 81 15 transform transform NOUN american_scientific_journal-3517 81 16 and and CCONJ american_scientific_journal-3517 81 17 its its PRON american_scientific_journal-3517 81 18 properties property NOUN american_scientific_journal-3517 81 19 in in ADP american_scientific_journal-3517 81 20 2d 2d NUM american_scientific_journal-3517 81 21 lct lct PROPN american_scientific_journal-3517 81 22 domain domain NOUN american_scientific_journal-3517 81 23 , , PUNCT american_scientific_journal-3517 81 24 signal signal NOUN american_scientific_journal-3517 81 25 processing processing NOUN american_scientific_journal-3517 81 26 89 89 NUM american_scientific_journal-3517 81 27 ( ( PUNCT american_scientific_journal-3517 81 28 2009 2009 NUM american_scientific_journal-3517 81 29 ) ) PUNCT american_scientific_journal-3517 81 30 1395–1402 1395–1402 NUM american_scientific_journal-3517 81 31 . . PUNCT american_scientific_journal-3517 82 1 [ [ X american_scientific_journal-3517 82 2 11 11 NUM american_scientific_journal-3517 82 3 ] ] PUNCT american_scientific_journal-3517 82 4 s. s. PROPN american_scientific_journal-3517 82 5 sarkar sarkar PROPN american_scientific_journal-3517 82 6 , , PUNCT american_scientific_journal-3517 82 7 k. k. PROPN american_scientific_journal-3517 82 8 mukherjee mukherjee PROPN american_scientific_journal-3517 82 9 , , PUNCT american_scientific_journal-3517 82 10 a. a. NOUN american_scientific_journal-3517 82 11 ray ray PROPN american_scientific_journal-3517 82 12 , , PUNCT american_scientific_journal-3517 82 13 generalization generalization NOUN american_scientific_journal-3517 82 14 of of ADP american_scientific_journal-3517 82 15 hilbert hilbert PROPN american_scientific_journal-3517 82 16 transform transform NOUN american_scientific_journal-3517 82 17 for for ADP american_scientific_journal-3517 82 18 symbolic symbolic ADJ american_scientific_journal-3517 82 19 analysis analysis NOUN american_scientific_journal-3517 82 20 of of ADP american_scientific_journal-3517 82 21 noisy noisy ADJ american_scientific_journal-3517 82 22 signals signal NOUN american_scientific_journal-3517 82 23 , , PUNCT american_scientific_journal-3517 82 24 signal signal ADJ american_scientific_journal-3517 82 25 processing processing NOUN american_scientific_journal-3517 82 26 89 89 NUM american_scientific_journal-3517 82 27 ( ( PUNCT american_scientific_journal-3517 82 28 2009 2009 NUM american_scientific_journal-3517 82 29 ) ) PUNCT american_scientific_journal-3517 82 30 1245–1251 1245–1251 NUM american_scientific_journal-3517 82 31 . . PUNCT american_scientific_journal-3517 83 1 [ [ X american_scientific_journal-3517 83 2 12 12 NUM american_scientific_journal-3517 83 3 ] ] PUNCT american_scientific_journal-3517 83 4 j. j. PROPN american_scientific_journal-3517 83 5 ge ge PROPN american_scientific_journal-3517 83 6 , , PUNCT american_scientific_journal-3517 83 7 c. c. PROPN american_scientific_journal-3517 83 8 wang wang PROPN american_scientific_journal-3517 83 9 , , PUNCT american_scientific_journal-3517 83 10 x. x. PROPN american_scientific_journal-3517 83 11 zhu zhu PROPN american_scientific_journal-3517 83 12 , , PUNCT american_scientific_journal-3517 83 13 fractional fractional ADJ american_scientific_journal-3517 83 14 optical optical ADJ american_scientific_journal-3517 83 15 hilbert hilbert NOUN american_scientific_journal-3517 83 16 transform transform NOUN american_scientific_journal-3517 83 17 using use VERB american_scientific_journal-3517 83 18 phase phase NOUN american_scientific_journal-3517 83 19 shifted shift VERB american_scientific_journal-3517 83 20 fiber fiber NOUN american_scientific_journal-3517 83 21 bragg bragg PROPN american_scientific_journal-3517 83 22 gratings gratings PROPN american_scientific_journal-3517 83 23 , , PUNCT american_scientific_journal-3517 83 24 optics optic NOUN american_scientific_journal-3517 83 25 communications communication NOUN american_scientific_journal-3517 83 26 284 284 NUM american_scientific_journal-3517 83 27 ( ( PUNCT american_scientific_journal-3517 83 28 2011 2011 NUM american_scientific_journal-3517 83 29 ) ) PUNCT american_scientific_journal-3517 83 30 3251–3257 3251–3257 NOUN american_scientific_journal-3517 83 31 . . PUNCT american_scientific_journal-3517 84 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 PROPN american_scientific_journal-3517 85 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 PROPN american_scientific_journal-3517 86 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 PROPN american_scientific_journal-3517 87 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 PROPN american_scientific_journal-3517 88 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 PROPN american_scientific_journal-3517 89 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref12 PROPN american_scientific_journal-3517 90 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 PROPN american_scientific_journal-3517 91 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 PROPN american_scientific_journal-3517 92 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 PROPN american_scientific_journal-3517 93 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 PROPN american_scientific_journal-3517 94 1 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 http://refhub.elsevier.com/s0165-1684(13)00190-4/sbref3 PROPN american_scientific_journal-3517 94 2 american american ADJ american_scientific_journal-3517 94 3 scientific scientific ADJ american_scientific_journal-3517 94 4 research research NOUN american_scientific_journal-3517 94 5 journal journal NOUN american_scientific_journal-3517 94 6 for for ADP american_scientific_journal-3517 94 7 engineering engineering NOUN american_scientific_journal-3517 94 8 , , PUNCT american_scientific_journal-3517 94 9 technology technology NOUN american_scientific_journal-3517 94 10 , , PUNCT american_scientific_journal-3517 94 11 and and CCONJ american_scientific_journal-3517 94 12 sciences science NOUN american_scientific_journal-3517 94 13 ( ( PUNCT american_scientific_journal-3517 94 14 asrjets asrjet NOUN american_scientific_journal-3517 94 15 ) ) PUNCT american_scientific_journal-3517 94 16 ( ( PUNCT american_scientific_journal-3517 94 17 2017 2017 NUM american_scientific_journal-3517 94 18 ) ) PUNCT american_scientific_journal-3517 94 19 volume volume NOUN american_scientific_journal-3517 94 20 38 38 NUM american_scientific_journal-3517 94 21 , , PUNCT american_scientific_journal-3517 94 22 no no DET american_scientific_journal-3517 94 23 2 2 NUM american_scientific_journal-3517 94 24 , , PUNCT american_scientific_journal-3517 94 25 pp pp ADV american_scientific_journal-3517 94 26 90 90 NUM american_scientific_journal-3517 94 27 - - SYM american_scientific_journal-3517 94 28 97 97 NUM american_scientific_journal-3517 94 29 97 97 NUM american_scientific_journal-3517 95 1 [ [ PUNCT american_scientific_journal-3517 95 2 13 13 NUM american_scientific_journal-3517 95 3 ] ] X american_scientific_journal-3517 95 4 k.n k.n PROPN american_scientific_journal-3517 95 5 . . PROPN american_scientific_journal-3517 95 6 chaudhury chaudhury PROPN american_scientific_journal-3517 95 7 , , PUNCT american_scientific_journal-3517 95 8 m. m. NOUN american_scientific_journal-3517 95 9 unser unser PROPN american_scientific_journal-3517 95 10 , , PUNCT american_scientific_journal-3517 95 11 on on ADP american_scientific_journal-3517 95 12 the the DET american_scientific_journal-3517 95 13 shiftability shiftability NOUN american_scientific_journal-3517 95 14 of of ADP american_scientific_journal-3517 95 15 dual dual ADJ american_scientific_journal-3517 95 16 - - PUNCT american_scientific_journal-3517 95 17 tree tree NOUN american_scientific_journal-3517 95 18 complex complex ADJ american_scientific_journal-3517 95 19 wavelet wavelet NOUN american_scientific_journal-3517 95 20 transforms transform VERB american_scientific_journal-3517 95 21 , , PUNCT american_scientific_journal-3517 95 22 ieee ieee NOUN american_scientific_journal-3517 95 23 transactions transaction NOUN american_scientific_journal-3517 95 24 on on ADP american_scientific_journal-3517 95 25 signal signal ADJ american_scientific_journal-3517 95 26 processing process VERB american_scientific_journal-3517 95 27 58 58 NUM american_scientific_journal-3517 95 28 ( ( PUNCT american_scientific_journal-3517 95 29 2010 2010 NUM american_scientific_journal-3517 95 30 ) ) PUNCT american_scientific_journal-3517 95 31 221–232 221–232 NUM american_scientific_journal-3517 95 32 . . PUNCT american_scientific_journal-3517 96 1 [ [ X american_scientific_journal-3517 96 2 14 14 NUM american_scientific_journal-3517 96 3 ] ] PUNCT american_scientific_journal-3517 96 4 a. a. NOUN american_scientific_journal-3517 96 5 sheikh sheikh PROPN american_scientific_journal-3517 96 6 , , PUNCT american_scientific_journal-3517 96 7 a. a. NOUN american_scientific_journal-3517 96 8 gudadhe gudadhe PROPN american_scientific_journal-3517 96 9 , , PUNCT american_scientific_journal-3517 96 10 on on ADP american_scientific_journal-3517 96 11 the the DET american_scientific_journal-3517 96 12 relationship relationship NOUN american_scientific_journal-3517 96 13 between between ADP american_scientific_journal-3517 96 14 genialized genialize VERB american_scientific_journal-3517 96 15 hilbert hilbert NOUN american_scientific_journal-3517 96 16 transform transform VERB american_scientific_journal-3517 96 17 with with ADP american_scientific_journal-3517 96 18 some some DET american_scientific_journal-3517 96 19 classical classical ADJ american_scientific_journal-3517 96 20 transforms transform NOUN american_scientific_journal-3517 96 21 , , PUNCT american_scientific_journal-3517 96 22 international international ADJ american_scientific_journal-3517 96 23 journal journal NOUN american_scientific_journal-3517 96 24 of of ADP american_scientific_journal-3517 96 25 scientific scientific ADJ american_scientific_journal-3517 96 26 and and CCONJ american_scientific_journal-3517 96 27 innovative innovative ADJ american_scientific_journal-3517 96 28 mathematical mathematical ADJ american_scientific_journal-3517 96 29 research research NOUN american_scientific_journal-3517 96 30 3(11 3(11 NUM american_scientific_journal-3517 96 31 ) ) PUNCT american_scientific_journal-3517 96 32 ( ( PUNCT american_scientific_journal-3517 96 33 2015 2015 NUM american_scientific_journal-3517 96 34 ) ) PUNCT american_scientific_journal-3517 96 35 3643 3643 NUM american_scientific_journal-3517 96 36 . . PUNCT american_scientific_journal-3517 97 1 [ [ X american_scientific_journal-3517 97 2 15 15 NUM american_scientific_journal-3517 97 3 ] ] X american_scientific_journal-3517 97 4 a. a. NOUN american_scientific_journal-3517 97 5 sheikh sheikh PROPN american_scientific_journal-3517 97 6 , , PUNCT american_scientific_journal-3517 97 7 a. a. NOUN american_scientific_journal-3517 97 8 gudadhe gudadhe PROPN american_scientific_journal-3517 97 9 , , PUNCT american_scientific_journal-3517 97 10 analyticity analyticity NOUN american_scientific_journal-3517 97 11 and and CCONJ american_scientific_journal-3517 97 12 operational operational ADJ american_scientific_journal-3517 97 13 transform transform NOUN american_scientific_journal-3517 97 14 on on ADP american_scientific_journal-3517 97 15 generalized generalized ADJ american_scientific_journal-3517 97 16 fractional fractional ADJ american_scientific_journal-3517 97 17 hilbert hilbert NOUN american_scientific_journal-3517 97 18 transform transform NOUN american_scientific_journal-3517 97 19 , , PUNCT american_scientific_journal-3517 97 20 vidarbha vidarbha PROPN american_scientific_journal-3517 97 21 journal journal PROPN american_scientific_journal-3517 97 22 of of ADP american_scientific_journal-3517 97 23 science science PROPN american_scientific_journal-3517 97 24 11(5 11(5 NUM american_scientific_journal-3517 97 25 ) ) PUNCT american_scientific_journal-3517 97 26 ( ( PUNCT american_scientific_journal-3517 97 27 2015 2015 NUM american_scientific_journal-3517 97 28 ) ) PUNCT american_scientific_journal-3517 97 29 47 47 NUM american_scientific_journal-3517 97 30 - - SYM american_scientific_journal-3517 97 31 51 51 NUM american_scientific_journal-3517 97 32 . . PUNCT american_scientific_journal-3517 98 1 [ [ X american_scientific_journal-3517 98 2 16 16 NUM american_scientific_journal-3517 98 3 ] ] PUNCT american_scientific_journal-3517 98 4 a. a. NOUN american_scientific_journal-3517 98 5 peng peng PROPN american_scientific_journal-3517 98 6 , , PUNCT american_scientific_journal-3517 98 7 y. y. PROPN american_scientific_journal-3517 98 8 wang wang PROPN american_scientific_journal-3517 98 9 , , PUNCT american_scientific_journal-3517 98 10 x. x. PROPN american_scientific_journal-3517 98 11 zuo zuo PROPN american_scientific_journal-3517 98 12 , , PUNCT american_scientific_journal-3517 98 13 l. l. PROPN american_scientific_journal-3517 98 14 gong gong PROPN american_scientific_journal-3517 98 15 , , PUNCT american_scientific_journal-3517 98 16 properties property NOUN american_scientific_journal-3517 98 17 of of ADP american_scientific_journal-3517 98 18 fractional fractional ADJ american_scientific_journal-3517 98 19 mellin mellin PROPN american_scientific_journal-3517 98 20 transform transform NOUN american_scientific_journal-3517 98 21 , , PUNCT american_scientific_journal-3517 98 22 advances advance NOUN american_scientific_journal-3517 98 23 in in ADP american_scientific_journal-3517 98 24 information information NOUN american_scientific_journal-3517 98 25 sciences sciences PROPN american_scientific_journal-3517 98 26 and and CCONJ american_scientific_journal-3517 98 27 services service NOUN american_scientific_journal-3517 98 28 sciences science NOUN american_scientific_journal-3517 98 29 5(11 5(11 NUM american_scientific_journal-3517 98 30 ) ) PUNCT american_scientific_journal-3517 98 31 ( ( PUNCT american_scientific_journal-3517 98 32 2013 2013 NUM american_scientific_journal-3517 98 33 ) ) PUNCT american_scientific_journal-3517 98 34 90 90 NUM american_scientific_journal-3517 98 35 - - SYM american_scientific_journal-3517 98 36 96 96 NUM american_scientific_journal-3517 98 37 . . PUNCT american_scientific_journal-3517 99 1 [ [ X american_scientific_journal-3517 99 2 17 17 NUM american_scientific_journal-3517 99 3 ] ] PUNCT american_scientific_journal-3517 99 4 j. j. PROPN american_scientific_journal-3517 99 5 a. a. PROPN american_scientific_journal-3517 99 6 davis davis PROPN american_scientific_journal-3517 99 7 , , PUNCT american_scientific_journal-3517 99 8 d.e d.e PROPN american_scientific_journal-3517 99 9 . . PROPN american_scientific_journal-3517 99 10 mcnamra mcnamra PROPN american_scientific_journal-3517 99 11 , , PUNCT american_scientific_journal-3517 99 12 d. d. PROPN american_scientific_journal-3517 99 13 m m PROPN american_scientific_journal-3517 99 14 cottrell cottrell PROPN american_scientific_journal-3517 99 15 , , PUNCT american_scientific_journal-3517 99 16 analysis analysis NOUN american_scientific_journal-3517 99 17 of of ADP american_scientific_journal-3517 99 18 fractional fractional ADJ american_scientific_journal-3517 99 19 hilbert hilbert NOUN american_scientific_journal-3517 99 20 transform transform NOUN american_scientific_journal-3517 99 21 , , PUNCT american_scientific_journal-3517 99 22 applied apply VERB american_scientific_journal-3517 99 23 optics optic NOUN american_scientific_journal-3517 99 24 37(29 37(29 NUM american_scientific_journal-3517 99 25 ) ) PUNCT american_scientific_journal-3517 99 26 ( ( PUNCT american_scientific_journal-3517 99 27 1998 1998 NUM american_scientific_journal-3517 99 28 ) ) PUNCT american_scientific_journal-3517 99 29 6911 6911 NUM american_scientific_journal-3517 99 30 - - SYM american_scientific_journal-3517 99 31 6913 6913 NUM american_scientific_journal-3517 99 32 . . PUNCT american_scientific_journal-3517 100 1 [ [ X american_scientific_journal-3517 100 2 18 18 NUM american_scientific_journal-3517 100 3 ] ] PUNCT american_scientific_journal-3517 100 4 a. a. NOUN american_scientific_journal-3517 100 5 sheikh sheikh PROPN american_scientific_journal-3517 100 6 , , PUNCT american_scientific_journal-3517 100 7 a. a. NOUN american_scientific_journal-3517 100 8 gudadhe gudadhe PROPN american_scientific_journal-3517 100 9 , , PUNCT american_scientific_journal-3517 100 10 on on ADP american_scientific_journal-3517 100 11 relationship relationship NOUN american_scientific_journal-3517 100 12 between between ADP american_scientific_journal-3517 100 13 genialized genialize VERB american_scientific_journal-3517 100 14 hilbert hilbert PROPN american_scientific_journal-3517 100 15 transform transform NOUN american_scientific_journal-3517 100 16 with with ADP american_scientific_journal-3517 100 17 other other ADJ american_scientific_journal-3517 100 18 transforms transform NOUN american_scientific_journal-3517 100 19 of of ADP american_scientific_journal-3517 100 20 fractional fractional ADJ american_scientific_journal-3517 100 21 order order NOUN american_scientific_journal-3517 100 22 , , PUNCT american_scientific_journal-3517 100 23 vidarbha vidarbha PROPN american_scientific_journal-3517 100 24 journal journal PROPN american_scientific_journal-3517 100 25 of of ADP american_scientific_journal-3517 100 26 science science NOUN american_scientific_journal-3517 100 27 11(01 11(01 NUM american_scientific_journal-3517 100 28 ) ) PUNCT american_scientific_journal-3517 100 29 ( ( PUNCT american_scientific_journal-3517 100 30 2016 2016 NUM american_scientific_journal-3517 100 31 ) ) PUNCT american_scientific_journal-3517 100 32 25 25 NUM american_scientific_journal-3517 100 33 - - SYM american_scientific_journal-3517 100 34 34 34 NUM american_scientific_journal-3517 100 35 . . PUNCT american_scientific_journal-3517 101 1 [ [ X american_scientific_journal-3517 101 2 19 19 NUM american_scientific_journal-3517 101 3 ] ] X american_scientific_journal-3517 101 4 r.a r.a PROPN american_scientific_journal-3517 101 5 . . PROPN american_scientific_journal-3517 101 6 treumann treumann PROPN american_scientific_journal-3517 101 7 , , PUNCT american_scientific_journal-3517 101 8 w. w. PROPN american_scientific_journal-3517 101 9 baumjohann baumjohann PROPN american_scientific_journal-3517 101 10 , , PUNCT american_scientific_journal-3517 101 11 fractional fractional ADJ american_scientific_journal-3517 101 12 laplace laplace NOUN american_scientific_journal-3517 101 13 transforms transform VERB american_scientific_journal-3517 101 14 a a DET american_scientific_journal-3517 101 15 perspective perspective NOUN american_scientific_journal-3517 101 16 , , PUNCT american_scientific_journal-3517 101 17 frontiers frontier NOUN american_scientific_journal-3517 101 18 physics physics NOUN american_scientific_journal-3517 101 19 2(2014 2(2014 NUM american_scientific_journal-3517 101 20 ) ) PUNCT american_scientific_journal-3517 101 21 1 1 NUM american_scientific_journal-3517 101 22 - - SYM american_scientific_journal-3517 101 23 4 4 NUM american_scientific_journal-3517 101 24 . . PUNCT american_scientific_journal-3517 102 1 [ [ X american_scientific_journal-3517 102 2 20 20 NUM american_scientific_journal-3517 102 3 ] ] X american_scientific_journal-3517 102 4 b.j b.j PROPN american_scientific_journal-3517 102 5 . . PROPN american_scientific_journal-3517 102 6 west west PROPN american_scientific_journal-3517 102 7 , , PUNCT american_scientific_journal-3517 102 8 m.bologna m.bologna PROPN american_scientific_journal-3517 102 9 , , PUNCT american_scientific_journal-3517 102 10 p. p. NOUN american_scientific_journal-3517 102 11 grigolini grigolini PROPN american_scientific_journal-3517 102 12 fractional fractional ADJ american_scientific_journal-3517 102 13 laplace laplace NOUN american_scientific_journal-3517 102 14 transforms transform VERB american_scientific_journal-3517 102 15 . . PUNCT american_scientific_journal-3517 103 1 in in ADP american_scientific_journal-3517 103 2 : : PUNCT american_scientific_journal-3517 103 3 physics physics NOUN american_scientific_journal-3517 103 4 of of ADP american_scientific_journal-3517 103 5 fractal fractal PROPN american_scientific_journal-3517 103 6 operators operator NOUN american_scientific_journal-3517 103 7 . . PUNCT american_scientific_journal-3517 104 1 institute institute PROPN american_scientific_journal-3517 104 2 for for ADP american_scientific_journal-3517 104 3 nonlinear nonlinear ADJ american_scientific_journal-3517 104 4 science science NOUN american_scientific_journal-3517 104 5 . . PUNCT american_scientific_journal-3517 105 1 springer springer NOUN american_scientific_journal-3517 105 2 , , PUNCT american_scientific_journal-3517 105 3 new new PROPN american_scientific_journal-3517 105 4 york york PROPN american_scientific_journal-3517 105 5 , , PUNCT american_scientific_journal-3517 105 6 ny ny PROPN american_scientific_journal-3517 105 7 ( ( PUNCT american_scientific_journal-3517 105 8 2003 2003 NUM american_scientific_journal-3517 105 9 ) ) PUNCT american_scientific_journal-3517 106 1 [ [ X american_scientific_journal-3517 106 2 21 21 NUM american_scientific_journal-3517 106 3 ] ] X american_scientific_journal-3517 106 4 v. v. ADP american_scientific_journal-3517 106 5 namias namias PROPN american_scientific_journal-3517 106 6 , , PUNCT american_scientific_journal-3517 106 7 β€œ " PUNCT american_scientific_journal-3517 106 8 the the DET american_scientific_journal-3517 106 9 fractional fractional ADJ american_scientific_journal-3517 106 10 order order NOUN american_scientific_journal-3517 106 11 fourier fourier NOUN american_scientific_journal-3517 106 12 transform transform NOUN american_scientific_journal-3517 106 13 and and CCONJ american_scientific_journal-3517 106 14 its its PRON american_scientific_journal-3517 106 15 application application NOUN american_scientific_journal-3517 106 16 to to ADP american_scientific_journal-3517 106 17 quantum quantum ADJ american_scientific_journal-3517 106 18 mechanics mechanic NOUN american_scientific_journal-3517 106 19 ” " PUNCT american_scientific_journal-3517 106 20 , , PUNCT american_scientific_journal-3517 106 21 i i PRON american_scientific_journal-3517 106 22 m m VERB american_scientific_journal-3517 106 23 a a DET american_scientific_journal-3517 106 24 journal journal NOUN american_scientific_journal-3517 106 25 of of ADP american_scientific_journal-3517 106 26 applied apply VERB american_scientific_journal-3517 106 27 mathematics mathematic NOUN american_scientific_journal-3517 106 28 , , PUNCT american_scientific_journal-3517 106 29 25(3 25(3 NUM american_scientific_journal-3517 106 30 ) ) PUNCT american_scientific_journal-3517 106 31 ( ( PUNCT american_scientific_journal-3517 106 32 1980 1980 NUM american_scientific_journal-3517 106 33 ) ) PUNCT american_scientific_journal-3517 106 34 241 241 NUM american_scientific_journal-3517 106 35 - - SYM american_scientific_journal-3517 106 36 265 265 NUM american_scientific_journal-3517 106 37 [ [ SYM american_scientific_journal-3517 106 38 22 22 NUM american_scientific_journal-3517 106 39 ] ] X american_scientific_journal-3517 106 40 l.b l.b PROPN american_scientific_journal-3517 106 41 . . PROPN american_scientific_journal-3517 106 42 almeida almeida PROPN american_scientific_journal-3517 106 43 , , PUNCT american_scientific_journal-3517 106 44 β€œ " PUNCT american_scientific_journal-3517 106 45 the the DET american_scientific_journal-3517 106 46 fractional fractional ADJ american_scientific_journal-3517 106 47 fourier fourier NOUN american_scientific_journal-3517 106 48 transform transform NOUN american_scientific_journal-3517 106 49 and and CCONJ american_scientific_journal-3517 106 50 time time NOUN american_scientific_journal-3517 106 51 - - PUNCT american_scientific_journal-3517 106 52 frequency frequency NOUN american_scientific_journal-3517 106 53 representations representation NOUN american_scientific_journal-3517 106 54 ” " PUNCT american_scientific_journal-3517 106 55 , , PUNCT american_scientific_journal-3517 106 56 ieee ieee NOUN american_scientific_journal-3517 106 57 transactions transaction NOUN american_scientific_journal-3517 106 58 on on ADP american_scientific_journal-3517 106 59 signal signal ADJ american_scientific_journal-3517 106 60 processing processing NOUN american_scientific_journal-3517 106 61 , , PUNCT american_scientific_journal-3517 106 62 42(11 42(11 NUM american_scientific_journal-3517 106 63 ) ) PUNCT american_scientific_journal-3517 106 64 ( ( PUNCT american_scientific_journal-3517 106 65 1994 1994 NUM american_scientific_journal-3517 106 66 ) ) PUNCT american_scientific_journal-3517 106 67 3084–3091 3084–3091 NUM american_scientific_journal-3517 106 68 , , PUNCT american_scientific_journal-3517 106 69 [ [ X american_scientific_journal-3517 106 70 23 23 NUM american_scientific_journal-3517 106 71 ] ] X american_scientific_journal-3517 106 72 s. s. PROPN american_scientific_journal-3517 106 73 iqbal iqbal PROPN american_scientific_journal-3517 106 74 , , PUNCT american_scientific_journal-3517 106 75 et et PROPN american_scientific_journal-3517 106 76 al al PROPN american_scientific_journal-3517 106 77 . . PROPN american_scientific_journal-3517 107 1 fractional fractional ADJ american_scientific_journal-3517 107 2 fourier fourier ADJ american_scientific_journal-3517 107 3 integral integral ADJ american_scientific_journal-3517 107 4 theorem theorem NOUN american_scientific_journal-3517 107 5 and and CCONJ american_scientific_journal-3517 107 6 fractional fractional ADJ american_scientific_journal-3517 107 7 fourier fourier NOUN american_scientific_journal-3517 107 8 , , PUNCT american_scientific_journal-3517 107 9 cosine cosine NOUN american_scientific_journal-3517 107 10 and and CCONJ american_scientific_journal-3517 107 11 sine sine VERB american_scientific_journal-3517 107 12 trans trans PROPN american_scientific_journal-3517 107 13 forms form NOUN american_scientific_journal-3517 107 14 , , PUNCT american_scientific_journal-3517 107 15 science science NOUN american_scientific_journal-3517 107 16 international international ADJ american_scientific_journal-3517 107 17 lahore lahore PROPN american_scientific_journal-3517 107 18 24(3 24(3 NUM american_scientific_journal-3517 107 19 ) ) PUNCT american_scientific_journal-3517 107 20 ( ( PUNCT american_scientific_journal-3517 107 21 2012 2012 NUM american_scientific_journal-3517 107 22 ) ) PUNCT american_scientific_journal-3517 107 23 233 233 NUM american_scientific_journal-3517 107 24 - - SYM american_scientific_journal-3517 107 25 238 238 NUM american_scientific_journal-3517 107 26 . . PUNCT american_scientific_journal-3517 108 1 [ [ X american_scientific_journal-3517 108 2 24 24 NUM american_scientific_journal-3517 108 3 ] ] X american_scientific_journal-3517 108 4 k.k k.k PROPN american_scientific_journal-3517 108 5 sharma sharma PROPN american_scientific_journal-3517 108 6 . . PUNCT american_scientific_journal-3517 108 7 fractional fractional ADJ american_scientific_journal-3517 108 8 laplace laplace NOUN american_scientific_journal-3517 108 9 transform transform NOUN american_scientific_journal-3517 108 10 , , PUNCT american_scientific_journal-3517 108 11 springer springer NOUN american_scientific_journal-3517 108 12 - - PUNCT american_scientific_journal-3517 108 13 verlag verlag PROPN american_scientific_journal-3517 108 14 london london PROPN american_scientific_journal-3517 108 15 limited limit VERB american_scientific_journal-3517 108 16 , , PUNCT american_scientific_journal-3517 108 17 4 4 NUM american_scientific_journal-3517 108 18 ( ( PUNCT american_scientific_journal-3517 108 19 2009 2009 NUM american_scientific_journal-3517 108 20 ) ) PUNCT american_scientific_journal-3517 108 21 377 377 NUM american_scientific_journal-3517 108 22 - - SYM american_scientific_journal-3517 108 23 379 379 NUM american_scientific_journal-3517 108 24 . . PUNCT