Mathematical Formulas Involving the Different Zeta Functions
In this chapter, a compendium of the formulas resulting from the zeta-regularization techniques, developed by the author and collaborators, is given. Although some of the original derivations are reproduced, the content of the chapter is mainly intended as an extensive table for practical use by the reader: the full derivations and arguments involved can be found in the accompanying bibliographical references. In particular, useful expressions are provided for the analytic continuation of Riemann, Hurwitz and Epstein zeta functions and generalizations of them, for their asymptotic expansions (including those for derivatives of Hurwitz’s, the zeta-function regularization theorem and its uses for multiple zeta-functions with arbitrary exponents and, in another section, the first immediate applications of the theorem. All this is followed by a very careful study of the analytic continuation of multiple series which terms are combinations involving arbitrary coefficients and exponents, a case that is very involved and has never before been treated properly in the mathematical literature. Of course this case always involves the elusive term that shows up in the correct application of the zeta-function regularization theorem, again an original result. Some mistakes which regretfully appeared some of in a few formulas of the original papers and previous versions have been corrected.
KeywordsAsymptotic Expansion Zeta Function Asymptotic Series Bernoulli Polynomial Regularization Theorem
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