id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejpam-4085	Reynolds, Robert; Stauffer, Allan	Double Integral Involving Logarithmic and Quotient Function with Powers Expressed in terms of the Lerch Function	2021	13	.pdf	application/pdf	3436	152	76	Use equation (12) and set m = 1/2, a = −1, q = 2 and simplify in terms of the Riemann zeta function using entry (2) in Tbale below (64:7) and entry (4) in table below (64:12:7) in [2] to get (23) ∫ ∞ 0 ∫ ∞ 0 √ y logk(−xy) √ x (x2 + 1)2 (y2 + 1)2 dxdy = 2k−5(iπ)k ( −8iπ ( 2k − 1 ) kζ(1− k) + ( 2k − 2 ) (k − 1)kζ(2− k)− 6π2 ( 2k+1 − 1 ) ζ(−k) ) Next apply L’Hopital’s rule as k → −1 and simplify. ∫ ∞ 0 ∫ ∞ 0 xm−1ym+ q 2 −1 logk(axy) (xq + 1)2 (yq + 1)2 dxdy = 1 2πi ∫ ∞ 0 ∫ ∞ 0 ∫ C aww−k−1xm+w−1ym+ q 2 +w−1 (xq + 1)2 (yq + 1)2 dwdxdy = 1 2πi ∫ C ∫ ∞ 0 ∫ ∞ 0 aww−k−1xm+w−1ym+ q 2 +w−1 (xq + 1)2 (yq + 1)2 dxdydw = 1 2πi ∫ C π2aww−k−1(m− q + w)(2m− q + 2w)	cache/ejpam-4085.pdf	txt/ejpam-4085.txt