Asymptotic expansions of the Hurwitz-Lerch zeta function |
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Authors: | Chelo Ferreira |
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Affiliation: | a Departamento de Matemática Aplicada, Universidad de Zaragoza, 50013 Zaragoza, Spain b Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain |
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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. |
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Keywords: | Hurwitz-Lerch zeta function Analytic continuation Asymptotic expansions |
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