The Lerch zeta function Ф(x, a, s) is defined by the series where x is real,0 < a ≤ 1, and σ = Re(s) > 1 if x is an integer and σ > 0 otherwise. In this paper we study the function J(s, a) = Ф(½, a, s). We use its integral representation to obtain the values of certain definite integrals; for example, we show that.

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Keywords Hurwitz zeta function, Integral representation, Lerch zeta function, Recurrence relations
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Journal Proceedings of the American Mathematical Society
Williams, K.S, & Nan-Yue, Z. (Zhang). (1993). Special values of the lerch zeta function and the evaluation of certain integrals. Proceedings of the American Mathematical Society, 119(1), 35–49. doi:10.1090/S0002-9939-1993-1172963-7