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Double integral of logarithm and exponential function expressed in terms of the Lerch function

Robert Reynolds, Allan Stauffer

TL;DR

The paper develops a contour-integral technique to evaluate a double integral of the form $\iint_{0}^{\infty}\int_{0}^{\infty} x^{m} y^{-m-1} e^{-p x - q y - x^{2}/(4y)} \log^{k}(a x/y)\,dx\,dy$, expressing the result in terms of the Lerch function $\Phi(z,s,v)$. By combining a generalized Cauchy formula with contour integration and analytic continuation of the Lerch function, the authors derive a main representation (Section 5) and a series of specific closed forms (Sections 6–18) for various parameter choices and logarithmic powers, including reductions to Prudnikov entries and new entries involving Hurwitz zeta, hypergeometric functions, and polylogarithms. A comprehensive summary table (Section 19) tabulates several double-integral cases and their closed forms, while the discussion (Section 20) notes numerical verification and outlines directions for future extensions. The work broadens the catalog of exact double-integral evaluations and demonstrates a versatile bridge between contour methods and special-function representations, with potential applications to moments, volumes, and error analyses in integral transforms.

Abstract

In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. The majority of the results in this work are new.

Double integral of logarithm and exponential function expressed in terms of the Lerch function

TL;DR

The paper develops a contour-integral technique to evaluate a double integral of the form , expressing the result in terms of the Lerch function . By combining a generalized Cauchy formula with contour integration and analytic continuation of the Lerch function, the authors derive a main representation (Section 5) and a series of specific closed forms (Sections 6–18) for various parameter choices and logarithmic powers, including reductions to Prudnikov entries and new entries involving Hurwitz zeta, hypergeometric functions, and polylogarithms. A comprehensive summary table (Section 19) tabulates several double-integral cases and their closed forms, while the discussion (Section 20) notes numerical verification and outlines directions for future extensions. The work broadens the catalog of exact double-integral evaluations and demonstrates a versatile bridge between contour methods and special-function representations, with potential applications to moments, volumes, and error analyses in integral transforms.

Abstract

In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus a table of integral pairs is given for interested readers. The majority of the results in this work are new.

Paper Structure

This paper contains 20 sections, 14 theorems, 22 equations, 1 table.

Table of Contents

  1. Introduction
  2. Definite integral of the contour integral
  3. The Lerch function
  4. Infinite sum of the contour integral
  5. Double integral in terms of the Lerch function
  6. Derivation of entry 3.1.3.48 in prud
  7. Derivation of new entry 3.1.3.59 in prud
  8. Derivation of new entry 3.1.3.60 in prud
  9. Derivation of new entry 3.1.3.61 in prud
  10. Derivation of new entry 3.1.3.62 in prud
  11. Derivation of new entry 3.1.3.63 in prud
  12. Derivation of new entry 3.1.3.64 in prud
  13. Derivation of new entry 3.1.3.65 in prud
  14. Derivation of new entry 3.1.3.66 in prud
  15. Derivation of new entry 3.1.3.67 in prud
  16. ...and 5 more sections

Key Result

Theorem 5.1

For $k,a,p,q \in\mathbb{C}, -1<Re(m)\leq -1/2, -1<Im(m)<-1/2$, where $\bar{r}=k!\bar{s}=-\frac{i \pi ^{k+1} 2^{k+m+1} q^{m/2} e^{-m \cosh ^{-1}\left(\frac{p}{\sqrt{q}}\right)+\frac{1}{2} i \pi (k+2 m)} }{\sqrt{p^2-q}}$

Theorems & Definitions (28)

  • Theorem 5.1
  • proof
  • Proposition 6.1
  • proof
  • Proposition 7.1
  • proof
  • Proposition 8.1
  • proof
  • Proposition 9.1
  • proof
  • ...and 18 more