Abstract :
The Hurwitz–Lerch zeta function Φ(z, s,a) is considered for large and small values of a ∈ C, and
for large values of z ∈ C, with |Arg(a)| < π, z /∈ [1,∞) and s ∈ C. This function is originally defined
as a power series in z, convergent for |z| < 1, s ∈ C and 1−a /∈ N. An integral representation is
obtained for Φ(z, s,a) which define the analytical continuation of the Hurwitz–Lerch zeta function
to the cut complex z-plane C \ [1,∞). 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.
2004 Elsevier Inc. All rights reserved.
