Double Integral of the Product of the Exponential of an Exponential Function and a Polynomial Expressed in Terms of the Lerch Function
In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-10-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/10/1962 |
| Summary: | In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mo>∞</mo></mrow><mo>∞</mo></msubsup><msubsup><mo>∫</mo><mrow><mo>−</mo><mo>∞</mo></mrow><mo>∞</mo></msubsup><msup><mi>e</mi><mrow><mi>m</mi><mi>x</mi><mo>−</mo><mi>m</mi><mi>y</mi><mo>−</mo><msup><mi>e</mi><mi>x</mi></msup><mo>−</mo><msup><mi>e</mi><mi>y</mi></msup><mo>+</mo><mi>y</mi></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>+</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>k</mi></msup><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mrow></semantics></math></inline-formula> in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. Almost all Lerch functions have an asymmetrical zero distribution. 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. All of the results in this work are new. |
|---|---|
| ISSN: | 2073-8994 |