Summary
The Poisson summation formula is employed to find the Laurent expansions of the Dirichlet seriesF(s, c) = Σ ∞ n = 0 exp[−(n + c)1/2 s] andG(s, c) = Σ ∞ n = 0 (−1)n exp[−(n + c)1/2 s] (0⩽c<1) abouts = 0. The Laurent expansions ofF(s, c) andG(s, c) are convergent respectively for 0 < |s| < ∞ and |s| < ∞, and define the analytic continuation of the Dirichlet series to the half-plane Res < 0.
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References
Abramowitz, M. andStegun, I.,Handbook of mathematical functions. Dover, New York, 1965.
Erdélyi, A., Magnus, W., Oberhettinger, F. andTricomi, F. G.,Higher transcendental functions. McGraw-Hill, New York, 1953.
Gradshteyn, I. S. andRyzhik, I. M.,Tables of integrals, series and products. Academic Press, New York, 1965.
Henrici, P.,Applied and computational complex variables. Vol. 2. Wiley, New York, 1977.
Holvorcem, P. R.,Asymptotic summation of Hermite series. J. Phys. A25 (1992), 909–924.
Jolley, L. B. W.,Summation of series. Dover, New York, 1961.
Saks, S. andZygmund, A.,Analytic functions. Elsevier, Amsterdam, 1971.
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Holvorcem, P.R. Laurent expansions for certain functions defined by Dirichlet series. Aequat. Math. 45, 62–69 (1993). https://doi.org/10.1007/BF01844425
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DOI: https://doi.org/10.1007/BF01844425