Asymptotic expansions of the Hurwitz-Lerch zeta function

The Hurwitz-Lerch zeta function Phi(z, s, a) is considered for large and small values of a is an element of C, and for large values of z is an element of C, with \Arg(a)\ < pi, z is not an element of [1, infinity) and s is an element of C. This function is originally defined as a power series in z, convergent for \z\ < 1, s is an element of C and 1 - a is not an element of N. An integral representation is obtained for Phi (z, s, a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C \ [1, infinity). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables. (C) 2004 Elsevier Inc. All rights reserved.