EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALS
A formula for the non-elementary integral \(\int e^{\lambda x^\alpha} dx\) where \(\alpha\) is real and greater or equal two, is obtained in terms of the confluent hypergeometric function \(_{1}F_1\) by expanding the integrand as a Taylor series. This result is verified by directly evaluating the ar...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
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
2017-12-01
|
| Series: | Ural Mathematical Journal |
| Subjects: | |
| Online Access: | https://umjuran.ru/index.php/umj/article/view/91 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849404229221875712 |
|---|---|
| author | Victor Nijimbere |
| author_facet | Victor Nijimbere |
| author_sort | Victor Nijimbere |
| collection | DOAJ |
| description | A formula for the non-elementary integral \(\int e^{\lambda x^\alpha} dx\) where \(\alpha\) is real and greater or equal two, is obtained in terms of the confluent hypergeometric function \(_{1}F_1\) by expanding the integrand as a Taylor series. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to \(\alpha=2\), using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function \(_{1}F_1\) and another one in terms of the hypergeometric function \(_1F_2\), are obtained for each of these integrals, \(\int\cosh(\lambda x^\alpha)dx\), \(\int\sinh(\lambda x^\alpha)dx\), \(\int \cos(\lambda x^\alpha)dx\) and \(\int\sin(\lambda x^\alpha)dx\), \(\lambda\in \mathbb{C},\alpha\ge2\). And the hypergeometric function \(_1F_2\) is expressed in terms of the confluent hypergeometric function \(_1F_1\). Some of the applications of the non-elementary integral \(\int e^{\lambda x^\alpha} dx, \alpha\ge 2\) such as the Gaussian distribution and the Maxwell-Bortsman distribution are given. |
| format | Article |
| id | doaj-art-9d2c69fb5b3e4cd9b419b4cdc04139f3 |
| institution | Kabale University |
| issn | 2414-3952 |
| language | English |
| publishDate | 2017-12-01 |
| 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-9d2c69fb5b3e4cd9b419b4cdc04139f32025-08-20T03:37:03ZengUral 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 Journal2414-39522017-12-013210.15826/umj.2017.2.01437EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALSVictor Nijimbere0School of Mathematics and Statistics, Carleton University, Ottawa, OntarioA formula for the non-elementary integral \(\int e^{\lambda x^\alpha} dx\) where \(\alpha\) is real and greater or equal two, is obtained in terms of the confluent hypergeometric function \(_{1}F_1\) by expanding the integrand as a Taylor series. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to \(\alpha=2\), using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function \(_{1}F_1\) and another one in terms of the hypergeometric function \(_1F_2\), are obtained for each of these integrals, \(\int\cosh(\lambda x^\alpha)dx\), \(\int\sinh(\lambda x^\alpha)dx\), \(\int \cos(\lambda x^\alpha)dx\) and \(\int\sin(\lambda x^\alpha)dx\), \(\lambda\in \mathbb{C},\alpha\ge2\). And the hypergeometric function \(_1F_2\) is expressed in terms of the confluent hypergeometric function \(_1F_1\). Some of the applications of the non-elementary integral \(\int e^{\lambda x^\alpha} dx, \alpha\ge 2\) such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.https://umjuran.ru/index.php/umj/article/view/91Non-elementary integral, Hypergeometric function, Confluent hypergeometric function, Asymptotic evaluation, Fundamental theorem of calculus, Gaussian, Maxwell-Bortsman distribution |
| spellingShingle | Victor Nijimbere EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALS Ural Mathematical Journal Non-elementary integral, Hypergeometric function, Confluent hypergeometric function, Asymptotic evaluation, Fundamental theorem of calculus, Gaussian, Maxwell-Bortsman distribution |
| title | EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALS |
| title_full | EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALS |
| title_fullStr | EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALS |
| title_full_unstemmed | EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALS |
| title_short | EVALUATION OF THE NON-ELEMENTARY INTEGRAL \(\int e^{\lambda x^\alpha}dx\), \(\alpha\ge 2\), AND OTHER RELATED INTEGRALS |
| title_sort | evaluation of the non elementary integral int e lambda x alpha dx alpha ge 2 and other related integrals |
| topic | Non-elementary integral, Hypergeometric function, Confluent hypergeometric function, Asymptotic evaluation, Fundamental theorem of calculus, Gaussian, Maxwell-Bortsman distribution |
| url | https://umjuran.ru/index.php/umj/article/view/91 |
| work_keys_str_mv | AT victornijimbere evaluationofthenonelementaryintegralintelambdaxalphadxalphage2andotherrelatedintegrals |