{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:13:48Z","timestamp":1760148828273,"version":"build-2065373602"},"reference-count":8,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,30]],"date-time":"2022-07-30T00:00:00Z","timestamp":1659139200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"NSERC Canada","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>A new three-dimensional integral containing f(x,y,z)Iv(x\u03b1) is derived where Iv(x\u03b1) is the Modified Bessel Function of the first kind and the integral is taken over the infinite cubic space 0&lt;x&lt;\u221e,0&lt;y&lt;\u221e,0&lt;z&lt;\u221e. The integral is not easily evaluated for complex ranges of the parameters. A representation in terms of the Hurwitz\u2013Lerch zeta function, polylogarithm function and Riemann zeta functions are evaluated. This representation yields triple integral representations in terms of fundamental constants that can be derived. Almost all Lerch functions have an asymmetrical zero distribution.<\/jats:p>","DOI":"10.3390\/sym14081573","type":"journal-article","created":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T23:49:27Z","timestamp":1659397767000},"page":"1573","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Non-Zero Order of an Extended Temme Integral"],"prefix":"10.3390","volume":"14","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, 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, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1090\/S0025-5718-1986-0856712-X","article-title":"A Double Integral Containing the Modified Bessel Function: Asymptotics and Computation","volume":"47","author":"Temme","year":"1986","journal-title":"Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"235","DOI":"10.12988\/imf.2020.91272","article-title":"A Method for Evaluating Definite Integrals in Terms of Special Functions with Examples","volume":"15","author":"Reynolds","year":"2020","journal-title":"Int. Math. Forum"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Brychkov, Y.A., Marichev, O.I., and Savischenko, N.V. (2019). Handbook of Mellin Transforms, Taylor & Francis, Chapman and Hall\/CRC.","DOI":"10.1201\/9780429434259"},{"key":"ref_4","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2000). Tables of Integrals, Series and Products, Academic Press. [6th ed.]."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Gelca, R., and Andreescu, T. (2007). Putnam and Beyond, Springer. [1st ed.].","DOI":"10.1007\/978-0-387-68445-1"},{"key":"ref_6","unstructured":"Olver, F.W.J., Lozier, D.W., Boisvert, R.F., and Clark, C.W. (2010). NIST Digital Library of Mathematical Functions, Cambridge University Press. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248 (2012a:33001)."},{"key":"ref_7","unstructured":"Laurincikas, A., and Garunkstis, R. (2013). The Lerch Zeta-Function, Springer Science & Business Media."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Oldham, K.B., Myland, J.C., and Spanier, J. (2009). An Atlas of Functions: With Equator, the Atlas Function Calculator, Springer. [2nd ed.].","DOI":"10.1007\/978-0-387-48807-3"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1573\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:59:41Z","timestamp":1760140781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/8\/1573"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,30]]},"references-count":8,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["sym14081573"],"URL":"https:\/\/doi.org\/10.3390\/sym14081573","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,7,30]]}}}