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Infinite integrals in terms of series

Robert Reynolds

TL;DR

The paper develops a robust contour-integral framework to express infinite integrals of products of generalized logarithms with rational and polynomial kernels as finite series in the Hurwitz-Lerch zeta function. By combining Mellin transforms, Cauchy-type contour formulas, and careful handling of singularities, it derives multiple contour representations (involving reciprocal Pochhammer and reciprocal polynomial forms) and uses them to obtain wide classes of generalized Malmsten–Schröder–Mellin-type integrals. The results unify and extend numerous entries from classical integral tables and connect to broader special-function theory, including Lerch transcendent, Bessel and hypergeometric functions, as well as $q$-Pochhammer symbols. The developed finite-series expressions facilitate explicit evaluations and offer a path to new identities and applications in analysis, number theory, and mathematical physics.

Abstract

In this work we derive and evaluate some infinite integrals involving the product of a generalized logarithm and polynomial functions in the denominator. These integrals are expressed in terms of finite series involving the Hurwitz-Lerch zeta function. We produce special cases of these integrals in terms of other special functions and fundamental constants.

Infinite integrals in terms of series

TL;DR

The paper develops a robust contour-integral framework to express infinite integrals of products of generalized logarithms with rational and polynomial kernels as finite series in the Hurwitz-Lerch zeta function. By combining Mellin transforms, Cauchy-type contour formulas, and careful handling of singularities, it derives multiple contour representations (involving reciprocal Pochhammer and reciprocal polynomial forms) and uses them to obtain wide classes of generalized Malmsten–Schröder–Mellin-type integrals. The results unify and extend numerous entries from classical integral tables and connect to broader special-function theory, including Lerch transcendent, Bessel and hypergeometric functions, as well as -Pochhammer symbols. The developed finite-series expressions facilitate explicit evaluations and offer a path to new identities and applications in analysis, number theory, and mathematical physics.

Abstract

In this work we derive and evaluate some infinite integrals involving the product of a generalized logarithm and polynomial functions in the denominator. These integrals are expressed in terms of finite series involving the Hurwitz-Lerch zeta function. We produce special cases of these integrals in terms of other special functions and fundamental constants.

Paper Structure

This paper contains 38 sections, 50 theorems, 206 equations, 6 figures.

Table of Contents

  1. Significance Statement
  2. Introduction
  3. Preliminaries
  4. The Hurwitz-Lerch zeta function
  5. First contour integral representation involving the Pochhammer symbol
  6. Second contour integral representation involving a generalized polynomial function
  7. Third contour integral representation
  8. Derivation of contour integral representations
  9. The contour integral involving the reciprocal Pochhammer symbol
  10. Left-hand side contour integral
  11. Right-hand side contour integral
  12. The contour integral involving the reciprocal polynomial function
  13. Left-hand side contour integral
  14. Right-hand side contour integral
  15. Generalized infinite logarithm integrals in terms of finite series
  16. ...and 23 more sections

Key Result

Proposition 3.1

The polylogarithm function [MathWorks, https://www.mathworks.com/help/symbolic/sym.polylog.html]; where $Re(s)>1,0,z\leq 1$.

Figures (6)

  • Figure 1: Plot of the integrand in (\ref{['eq:malm_poxh_1']})
  • Figure 2: Plot of the integrand in (\ref{['eq:poch_sing_1']})
  • Figure 3: Plot of the partial sums of equation (\ref{['eq:gregory']})
  • Figure 4: Plot of the $\frac{1}{\left(\pi ^2-\log ^2(x)\right) (x+1)_{n+1}}$
  • Figure 5: Plot of the $Re\left(\frac{\left(-\frac{x}{2};\frac{1}{2}\right)_{\infty } \log (\log (x))}{\sqrt{x} \left(-x;\frac{1}{2}\right)_{\infty }}\right)$
  • ...and 1 more figures

Theorems & Definitions (191)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • ...and 181 more