id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-4203	Reynolds, Robert; Stauffer, Allan 	Triple Integral involving the Product of the Logarithmic and Bessel Functions expressed in terms of the Lerch Function	2022	7	.pdf	application/pdf	2120	111	73	Introduction In this paper we derive the triple definite integral given by (1) ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 tmxv−my 1 2 (−m−v−1)Jv(t)e −bx2−cy logk ( at x √ y ) dxdydt where the parameters k, a, b, c, v,m are general complex numbers and Re(b) > 0, Re(c) > 0, Re(v) > 0, Re(m) < −1. The Hurwitz-Lerch zeta Function The Hurwitz-Lerch zeta function (25.14) in [3] has a series representation given by Φ(z, s, v) = ∞∑ n=0 (v + n)−szn (4) where |z|< 1, v 6= 0,−1, .. and is continued analytically by its integral representation given by Φ(z, s, v) = 1 Γ(s) ∫ ∞ 0 ts−1e−vt 1− ze−t dt = 1 Γ(s) ∫ ∞ 0 ts−1e−(v−1)t et − z dt (5) where Re(v) > 0, and either |z|≤ 1, z 6= 1, Re(s) > 0, or z = 1, Re(s) > 1. R. Reynolds, A. Stauffer / Eur.	cache/ejpam-4203.pdf	txt/ejpam-4203.txt