90 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 Β© Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ The Relationship of Generalized Fractional Hilbert Transform with Fractional Mellin and Fractional Laplace Transforms Laljahana, Saleem Iqbalb*, Farhana Sarwarc, Syed Mohsin Razad, Abdul Rehmane a,b,eDepartment of Mathematics, University of Balochistan, Quetta 87300, Pakistan cDepartment of Mathematics F.G.Girls Degree College, Madrissa Road , Quetta, Cantt, 87300, Pakistan dDepartmen of Physics, University of Balochistan, Quetta 87300, Pakistan aEmail: laljahanbaloch@gmail.com bEmail: saleemiqbal81@yahoo.com cEmail: f_saleem10@yahoo.com dEmail: smraza7@yahoo.com eEmail: abdul_maths@yahoo.com Abstract We have developed in this research paper, some of the fundamental relationship between generalized fractional Hilbert transform with fractional Mellin transform, fractional Laplace transform, fractional inverse Laplace transform.. The results are mathematically expressed. These results, however, need modelling and simulation with any specialized signal processing data. Keywords: Fractional Hilbert Transform; Fractional Mellin Transform; Fractional Laplace Transform; Fractional Fourier transform. 1. Introduction Fractional Hilbert transform(FRHT) is introduced by Lohmann and his colleagues [1]. They proved in their paper that the FRHT is the generalization of the Hilbert transform(HT). Their generalization is based on modifying the spatial filters and fractional Fourier plane for filters. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 2, pp 90-97 91 In continuation of the work [1], Davis and his colleagues [2] employed FRFT for edge detection and proved that as the fractional order is varied then different qualities of edge enhancement are obtained. In [3] discrete counterpart of the FRHT is discussed. Zayed [4] took the analytical signal (AS) formalism associated with the standard Fourier transform (FT) and provided a counterpart of it for the fractional Fourier transform (FRFT). The communication applications is discussed by using FRHT in [5] Fractional Hilbert transform is a special case of linear canonical transform (LCT) which has diverse applications such as in an image enhancement or compression by using the angle of rotation in the complex plane (t,w) of fractional Fourier transform (FRFT) on optical systems, edge deduction for propagation and delay times , beam flowing and indeed in signal processing [6-13]. Needless to mention, the integral transforms have significant applications in both Physics and Applied Mathematics. Akilahmad Sheikh and Alka Gudadhe [14,15] worked on relationships on generalized fractional Hilbert transform with some classical transforms and developed analytic theorems. The fractional Hilbert transform is a generalization of Hilbert transform and so is the case of fractional Fourier tansform (FRFT). The scale invariance property of fractional Mellin transform (FMT) is a very important tool to constructing two dimensional real images [16]. Analysis of Hilbert transforms with fractional Fourier transform (FRFT) [17-18] resulted into many fascinating results. The use of fractional Hilbert transforms and its extensions lead to analytical behavior of signal constructions. [9]. This is how amplitude modulation (AM,) frequency modulation (FM) or phase modulation(PM) can be exploited in Weiner filter for controlling the changes in phase signals. These fractional phase changes are the manifestations of the Fractional Hilbert transform and of the Morlet and Harlet wavelets for two super posed wave form. The same behavior is witnessed with Fourier transform [4]. The Fractional Laplace transform is also the genialized case of Laplace transform. Heaviside step function is a step forward to deal with Laplace transform because it deals in the denominator with a function with fractional exponent. Laplace and its corresponding fractional Laplace transform deals with Wigner distribution(WD), Wiener space(WS) the ambiguity fraction, the short time Fourier transform(SSTFT), speech processing, radar, image rotation, Confocal microscopy, etc. The aperiodic stable chaos with Lyapunov experiments in real time signals can be studied with fractional Laplace transform [19-20]. 2. Results and Discussions 2.1. The relationship between generalized fractional Hilbert transform with fractional Mellin transform The relation between Fractional Mellin transform and fractional Fourier is defined in [16] as 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(π‘₯π‘₯)] = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(𝑒𝑒𝑑𝑑)] = οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝑓𝑓(𝑒𝑒𝑑𝑑) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 (1) FRFT is the fractional Fourier transform and its definition and its applications are disused in [21-23]. Where FRMT is fractional Mellin transform and defined as American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 2, pp 90-97 92 𝐹𝐹𝑖𝑖(𝑒𝑒) = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹[𝑓𝑓(𝑖𝑖)] = οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 The generalize fractional Hilbert transform is defined in [14] as 𝐻𝐻𝑖𝑖[𝑓𝑓(π‘₯π‘₯)](𝑖𝑖) = π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑑𝑑2 πœ‹πœ‹ οΏ½ 𝑓𝑓(π‘₯π‘₯) 𝑖𝑖 βˆ’ π‘₯π‘₯ ∞ βˆ’βˆž 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 π‘₯π‘₯2𝑑𝑑π‘₯π‘₯ (2) 𝐻𝐻𝑖𝑖 �𝑒𝑒 𝑖𝑖 2(βˆ’π‘–π‘–2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)οΏ½ (𝑦𝑦) = π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 𝑒𝑒(βˆ’π‘–π‘–2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝐹𝐹 𝑖𝑖[𝑓𝑓(𝑖𝑖)] 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2𝑑𝑑𝑒𝑒 = π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž οΏ½οΏ½ 1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝑓𝑓(𝑒𝑒𝑑𝑑) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖�𝑑𝑑𝑒𝑒 = 𝑖𝑖 𝑖𝑖𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐2 𝑦𝑦2 πœ‹πœ‹ οΏ½1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 2πœ‹πœ‹ ∫ 𝑖𝑖𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑐𝑐2 π‘¦π‘¦βˆ’π‘–π‘– ∞ βˆ’βˆž �∫ 𝑓𝑓(𝑒𝑒𝑑𝑑)∞ βˆ’βˆž 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‘–οΏ½π‘‘π‘‘π‘’π‘’ By changing the order of integration = 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝑓𝑓(𝑒𝑒𝑑𝑑)𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑑𝑑2 ∞ βˆ’βˆž οΏ½ 1 πœ‹πœ‹ οΏ½ π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘– 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž 𝑑𝑑𝑒𝑒� 𝑑𝑑𝑖𝑖 (3) From [18] we have οΏ½ 𝑒𝑒𝑖𝑖𝑖𝑖𝑑𝑑 π‘₯π‘₯ βˆ’ 𝑖𝑖 ∞ βˆ’βˆž 𝑑𝑑𝑖𝑖 = βˆ’π‘–π‘– 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑖𝑖π‘₯π‘₯ (4) equation (3) can be written as = 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝑓𝑓(𝑒𝑒𝑑𝑑) ∞ βˆ’βˆž 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑑𝑑2(𝑖𝑖)𝑠𝑠𝑠𝑠𝑠𝑠 οΏ½ 𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 οΏ½ 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‘– =𝑖𝑖�1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 2πœ‹πœ‹ ∫ 𝑓𝑓(𝑒𝑒𝑑𝑑)∞ βˆ’βˆž 𝑠𝑠𝑠𝑠𝑠𝑠 οΏ½ 𝑑𝑑 𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖 οΏ½ π‘’π‘’βˆ’π‘–π‘–π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2(𝑑𝑑2+𝑦𝑦2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 𝐻𝐻𝑖𝑖{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)]}(𝑒𝑒) = 𝑖𝑖𝐹𝐹𝑖𝑖 �𝑆𝑆𝑠𝑠𝑠𝑠 οΏ½ 𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 οΏ½ 𝑓𝑓(𝑖𝑖)οΏ½ (5) Equation (4) is the relation between generalized fractional Hilbert transform and Fractional Mellin transform. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 2, pp 90-97 93 2.2. The relationship between generalized fractional inverse Hilbert transform with fractional Mellin transform Let 𝑠𝑠(𝑒𝑒) = ∫ 𝑓𝑓(𝑖𝑖)∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 ( 5) 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ 𝑠𝑠(𝑒𝑒) = οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2(𝑑𝑑2+𝑖𝑖2)𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 𝑒𝑒𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑖𝑖2οΏ½1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 2πœ‹πœ‹ 𝑠𝑠(𝑒𝑒) = 𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒) (6) π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦) = βˆ’ 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒) 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2𝑑𝑑𝑒𝑒 = βˆ’π‘’π‘’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 1 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 = βˆ’π‘’π‘’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 1 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž οΏ½οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖� 𝑑𝑑𝑒𝑒 𝑏𝑏𝑦𝑦 π‘–π‘–β„Žπ‘Žπ‘Žπ‘ π‘ π‘ π‘ π‘–π‘–π‘ π‘ π‘ π‘  π‘–π‘–β„Žπ‘’π‘’ π‘–π‘–π‘œπ‘œπ‘‘π‘‘π‘’π‘’π‘œπ‘œ 𝑖𝑖𝑓𝑓 π‘–π‘–π‘ π‘ π‘–π‘–π‘’π‘’π‘ π‘ π‘œπ‘œπ‘Žπ‘Žπ‘–π‘–π‘–π‘–π‘–π‘–π‘ π‘  π»π»βˆ’π‘–π‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦) = βˆ’π‘’π‘’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½1 βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 οΏ½ 1 πœ‹πœ‹ οΏ½ 1 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž 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‘’οΏ½ 𝑑𝑑𝑖𝑖 (7) We have from [18] ∫ 𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐 π‘₯π‘₯βˆ’π‘‘π‘‘ ∞ βˆ’βˆž 𝑑𝑑𝑖𝑖 = βˆ’π‘–π‘– 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑖𝑖π‘₯π‘₯ (8) Equation (7) can be written as = βˆ’π‘–π‘–π‘–π‘– 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑦𝑦2 πœ‹πœ‹ οΏ½1βˆ’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– 2πœ‹πœ‹ ∫ 𝑓𝑓(𝑖𝑖)∞ βˆ’βˆž 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 . 𝑖𝑖. 𝑠𝑠𝑠𝑠𝑠𝑠 οΏ½ 𝑑𝑑 𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖 οΏ½ 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‘–{𝐹𝐹𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒)}(𝑦𝑦) = βˆ’π‘–π‘–πΉπΉπ‘–π‘– �𝑆𝑆𝑠𝑠𝑠𝑠 οΏ½ 𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 οΏ½ 𝑓𝑓(𝑖𝑖)οΏ½ (𝑦𝑦) (9) Equation (9) is the relation between generalized fractional inverse Hilbert transform and Fractional Mellin transform. 2.3 Relationship between generalized fractional invers Hilbert transform with fractional Laplace transform American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 2, pp 90-97 94 Fractional Laplace transform is defined in [24] as 𝐿𝐿𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒) = π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 (10) Using equation (5) π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 𝑠𝑠(𝑒𝑒) = π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 𝑠𝑠(𝑒𝑒) = 𝐿𝐿𝑖𝑖[𝑓𝑓(𝑖𝑖)](𝑒𝑒) (11) Now consider π»π»βˆ’π‘–π‘–οΏ½π‘’π‘’π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–(𝑖𝑖2βˆ’π‘¦π‘¦2)𝐿𝐿𝑖𝑖[𝐹𝐹(𝑖𝑖)](𝑒𝑒)οΏ½(𝑦𝑦) = βˆ’ 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖(𝑖𝑖2βˆ’π‘¦π‘¦2) 𝐿𝐿 𝑖𝑖[𝐹𝐹(𝑖𝑖)](𝑒𝑒) 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2𝑑𝑑𝑒𝑒 = βˆ’ π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 1 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž 𝑒𝑒𝑖𝑖 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑖𝑖2οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– 𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 = βˆ’ π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– οΏ½ 1 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž 𝑠𝑠(𝑒𝑒)𝑑𝑑𝑒𝑒 = βˆ’ π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– οΏ½ 1 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž οΏ½οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘’π‘’ 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖� 𝑑𝑑𝑒𝑒 (12) Interchanging the integration order = βˆ’π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 οΏ½οΏ½ 1 𝑦𝑦 βˆ’ 𝑒𝑒 ∞ βˆ’βˆž π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘’π‘’οΏ½ 𝑑𝑑𝑖𝑖 (13) = βˆ’π‘’π‘’βˆ’π‘–π‘– 𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½ 1 2πœ‹πœ‹π‘ π‘ π‘–π‘–π‘ π‘ π‘–π‘– οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠 οΏ½ 𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 οΏ½ π‘’π‘’βˆ’π‘¦π‘¦π‘‘π‘‘π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘– (14) π»π»βˆ’π‘–π‘–οΏ½πΏπΏπ‘–π‘–οΏ½πΉπΉ(𝑖𝑖)𝑒𝑒𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑖𝑖2οΏ½(𝑒𝑒)οΏ½(𝑦𝑦) = βˆ’πΏπΏπ‘–π‘– οΏ½οΏ½ 𝑓𝑓(𝑖𝑖) ∞ βˆ’βˆž 𝑠𝑠𝑠𝑠𝑠𝑠 οΏ½ 𝑖𝑖 𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖 οΏ½οΏ½ (𝑦𝑦) (15) Equation (15) is the relation between generalized fractional Hilbert transform and Fractional Laplace transform. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 2, pp 90-97 95 2.4 The relationship between generalized fractional Hilbert transform with fractional inverse Laplace transforms The fraction inverse Laplace transform is defined in [24] 𝑓𝑓(𝑖𝑖) = οΏ½1 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 οΏ½ 𝐹𝐹 𝑖𝑖+∞ π‘–π‘–βˆ’βˆž (𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑖𝑖 4𝑑𝑑 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 π»π»π‘–π‘–οΏ½π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖)οΏ½(𝑦𝑦) = π‘’π‘’βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) 𝑦𝑦 βˆ’ 𝑖𝑖 ∞ βˆ’βˆž 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 = π‘’π‘’βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑦𝑦 βˆ’ 𝑖𝑖 ∞ βˆ’βˆž 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 οΏ½οΏ½ 1 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 οΏ½ 𝐹𝐹𝐿𝐿.𝑖𝑖 𝑖𝑖+∞ π‘–π‘–βˆ’βˆž (𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑖𝑖𝑖𝑖2 4 𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒�𝑑𝑑𝑖𝑖 = π‘’π‘’βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½1 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝐹𝐹𝐿𝐿.𝑖𝑖 𝑖𝑖+∞ π‘–π‘–βˆ’βˆž (𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑖𝑖 4𝑑𝑑 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 οΏ½ 1 πœ‹πœ‹ οΏ½ 𝑒𝑒𝑖𝑖𝑑𝑑 𝑦𝑦 βˆ’ 𝑖𝑖 ∞ βˆ’βˆž 𝑑𝑑𝑖𝑖� 𝑑𝑑𝑒𝑒 = π‘’π‘’βˆ’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½1 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝐹𝐹𝐿𝐿.𝑖𝑖 𝑖𝑖+∞ π‘–π‘–βˆ’βˆž (𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑖𝑖 4𝑑𝑑 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 . 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 π»π»π‘–π‘–οΏ½π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖)οΏ½(𝑦𝑦) = πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑦𝑦) On the relationship between generalized inverse fractional Hilbert transform with fractional inverse Laplace transform π»π»βˆ’π‘–π‘–οΏ½π‘’π‘’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖)οΏ½(𝑦𝑦) = βˆ’π‘’π‘’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖) 𝑦𝑦 βˆ’ 𝑖𝑖 ∞ βˆ’βˆž 𝑒𝑒 βˆ’π‘–π‘– 2 𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖𝑑𝑑𝑖𝑖 = βˆ’π‘’π‘’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2 πœ‹πœ‹ οΏ½ 𝑒𝑒𝑖𝑖𝑑𝑑2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 𝑦𝑦 βˆ’ 𝑖𝑖 ∞ βˆ’βˆž π‘’π‘’βˆ’ 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 οΏ½οΏ½ 1 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ 𝑒𝑒 𝑖𝑖 2𝑑𝑑 2𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 οΏ½ 𝐹𝐹𝐿𝐿.𝑖𝑖 𝑖𝑖+∞ π‘–π‘–βˆ’βˆž (𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑖𝑖𝑖𝑖2 4 𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒�𝑑𝑑𝑖𝑖 = βˆ’π‘’π‘’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½1 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝐹𝐹𝐿𝐿.𝑖𝑖 𝑖𝑖+∞ π‘–π‘–βˆ’βˆž (𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑖𝑖 4𝑑𝑑 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 οΏ½ 1 πœ‹πœ‹ οΏ½ 𝑒𝑒𝑖𝑖𝑑𝑑 𝑦𝑦 βˆ’ 𝑖𝑖 ∞ βˆ’βˆž 𝑑𝑑𝑖𝑖� 𝑑𝑑𝑒𝑒 = βˆ’π‘’π‘’ 𝑖𝑖𝑖𝑖𝑖𝑖𝑑𝑑𝑖𝑖 2 𝑦𝑦2οΏ½1 + 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2πœ‹πœ‹ οΏ½ 𝐹𝐹𝐿𝐿.𝑖𝑖 𝑖𝑖+∞ π‘–π‘–βˆ’βˆž (𝑒𝑒𝑠𝑠𝑖𝑖𝑠𝑠𝑖𝑖)π‘’π‘’βˆ’ 𝑖𝑖 4𝑑𝑑 2𝑖𝑖𝑖𝑖𝑠𝑠2𝑖𝑖 . 𝑠𝑠𝑠𝑠𝑠𝑠(𝑒𝑒)𝑒𝑒𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒 π»π»π‘–π‘–οΏ½π‘’π‘’βˆ’π‘–π‘–π‘‘π‘‘2π‘–π‘–π‘–π‘–π‘‘π‘‘π‘–π‘–πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑖𝑖)οΏ½(𝑦𝑦) = βˆ’πΏπΏβˆ’π‘–π‘–[𝑓𝑓(𝑒𝑒)](𝑦𝑦) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2017) Volume 38, No 2, pp 90-97 96 3. 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