In the paper, a generalization of the asymptotic expansions obtained by M. Katsurada [Proc. Japan Acad. 74, No. 10, 167–170 (1998; Zbl 0937.11035)] and D. Klusch [J. Math. Anal. Appl. 170, No. 2, 513–523 (1992; Zbl 0763.11036)] for the Lipschitz-Lerch zeta function
to the Hurwitz-Lerch zeta function
is presented. Note that . First, using an integral formula for the Hurwitz-Lerch zeta function
given in [H. M. Srivastava and J. Choi, Series associated with the zeta and related functions. Dordrecht: Kluwer Academic Publishers (2001; Zbl 1014.33001)], the authors obtain an integral representation which gives the analytical continuation of the function to the region if , and if , . From this they deduce three complete asymptotic expansions for either large or small and large with error bounds. Moreover, the numerical examples for these bounds are presented.