{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:27:21Z","timestamp":1760236041015,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,18]],"date-time":"2021-10-18T00:00:00Z","timestamp":1634515200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council","doi-asserted-by":"publisher","award":["504070"],"award-info":[{"award-number":["504070"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by \u222b\u2212\u221e\u221e\u222b\u2212\u221e\u221eemx\u2212my\u2212ex\u2212ey+y(log(a)+x\u2212y)kdxdy 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.<\/jats:p>","DOI":"10.3390\/sym13101962","type":"journal-article","created":{"date-parts":[[2021,10,20]],"date-time":"2021-10-20T22:07:04Z","timestamp":1634767624000},"page":"1962","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Double Integral of the Product of the Exponential of an Exponential Function and a Polynomial Expressed in Terms of the Lerch Function"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4230-9925","authenticated-orcid":false,"given":"Robert","family":"Reynolds","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7252-5004","authenticated-orcid":false,"given":"Allan","family":"Stauffer","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,18]]},"reference":[{"key":"ref_1","unstructured":"Nielsen, N. (1906). Handbuch der Theorie der Gammafunktion, Teubner."},{"key":"ref_2","unstructured":"Max, B., and Emil, W. (1980). Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Pergamon Press. [6th ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Neretin, Y.A. (2011). Lectures on Gaussian Integral Operators and Classical Groups, European Mathematical Society.","DOI":"10.4171\/080"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1103\/PhysRevLett.58.2718","article-title":"Scattering Properties of a Model Bicontinuous Structure with a Well Defined Length Scale","volume":"58","author":"Berk","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","unstructured":"Fourier, J. (1822). Th\u00e9orie Analytique de la Chaleur, F. Didot."},{"key":"ref_6","unstructured":"Titchmarsh, E.C. (1962). Introduction to the Theory of Fourier Integrals, Oxford at the Clarendon Press. 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[2nd ed.].","DOI":"10.1007\/978-0-387-48807-3"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/s11139-007-9102-0","article-title":"Double integrals and infinite products for some classical constants via analytic continuations of Lerch\u2019s transcendent","volume":"16","author":"Guillera","year":"2008","journal-title":"Ramanujan J."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Borwein, J., Bailey, D., and Girgensohn, R. (2004). Experimentation in Mathematics: Computational Paths to Discovery, A K Peters.","DOI":"10.1201\/9781439864197"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1962\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:17:07Z","timestamp":1760167027000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/10\/1962"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,18]]},"references-count":13,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["sym13101962"],"URL":"https:\/\/doi.org\/10.3390\/sym13101962","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,10,18]]}}}