id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-4700	Corcino, Roberto Bagsarsa; Corcino, Cristina; Casas, Karl Patrick; Elnar, Allan Roy; Maglasang, Gibson	Construction of Fourier Series Expansion of Apostol-Frobenius-Type Tangent and Genocchi Polynomials	2023	19	.pdf	application/pdf	5609	286	83	We then find the other poles of the function fn(t) as follows: λe2t − u = 0 λe2t = u λ 2t = log u λ + 2kπi tk := t = log (u λ )1/2 + kπi, for k ∈ Z By Cauchy Residue Theorem, we have 1 2πi ∫ CN fn(t)dt = Res(f(t), t = 0) Math, 16 (2) (2023), 1005-1023 1012 = u− 1 u 2n e−ξπix (−πi)n+1 [∫ ∞ 0 e 1 2 πix eπix − e−2t e−(2ξ+1)ttndt + (−1)n+1 ∫ ∞ 0 e 1 2 πix e2t − eiπx e(2ξ+1)ttndt ] = u− 1 u 2n e−ξπix (−πi)n+1 [∫ ∞ 0 e−(2ξ+1)t e 1 2 πix − e−2t− 1 2 πix tndt + (−1)n+1 ∫ ∞ 0 e(2ξ+1)t e2t− 1 2 πix − e 1 2 πix tndt ] = u− 1 u 2n e−ξπix (−πi)n+1 [∫ ∞ 0 e−(2ξ+1)t e 1 2 πix − e−2t− 1 2 πix · ( e2t − eπix )	cache/ejpam-4700.pdf	txt/ejpam-4700.txt