DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION
A family of generalized definite logarithmic integrals given by $$ \int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$ built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2021-07-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/321 |
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| author | Robert Reynolds Allan Stauffer |
| author_facet | Robert Reynolds Allan Stauffer |
| author_sort | Robert Reynolds |
| collection | DOAJ |
| description | A family of generalized definite logarithmic integrals given by $$
\int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$
built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general method as given in [6] to derive this integral. We then give a number of examples that can be derived from the general integral in terms of well known functions. |
| format | Article |
| id | doaj-art-34eacac67ec34a1a88dd314fe24e76d3 |
| institution | Kabale University |
| issn | 2414-3952 |
| language | English |
| publishDate | 2021-07-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-34eacac67ec34a1a88dd314fe24e76d32025-08-20T03:32:57ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522021-07-017110.15826/umj.2021.1.008119DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTIONRobert Reynolds0Allan Stauffer1Department of Mathematics and Statistics, York University, 4700 Keele Street, TorontoDepartment of Mathematics and Statistics, York University, 4700 Keele Street, TorontoA family of generalized definite logarithmic integrals given by $$ \int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$ built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general method as given in [6] to derive this integral. We then give a number of examples that can be derived from the general integral in terms of well known functions.https://umjuran.ru/index.php/umj/article/view/321entries of gradshteyn and ryzhik, lerch function, knuth's series |
| spellingShingle | Robert Reynolds Allan Stauffer DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION Ural Mathematical Journal entries of gradshteyn and ryzhik, lerch function, knuth's series |
| title | DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION |
| title_full | DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION |
| title_fullStr | DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION |
| title_full_unstemmed | DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION |
| title_short | DEFINITE INTEGRAL OF LOGARITHMIC FUNCTIONS AND POWERS IN TERMS OF THE LERCH FUNCTION |
| title_sort | definite integral of logarithmic functions and powers in terms of the lerch function |
| topic | entries of gradshteyn and ryzhik, lerch function, knuth's series |
| url | https://umjuran.ru/index.php/umj/article/view/321 |
| work_keys_str_mv | AT robertreynolds definiteintegraloflogarithmicfunctionsandpowersintermsofthelerchfunction AT allanstauffer definiteintegraloflogarithmicfunctionsandpowersintermsofthelerchfunction |