EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I
The non-elementary integrals \(\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le\beta+1\) and \(\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le2\beta+1\), where \(\{\beta,\alpha\}\i...
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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
2018-07-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/108 |
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| author | Victor Nijimbere |
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| author_sort | Victor Nijimbere |
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| description | The non-elementary integrals \(\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le\beta+1\) and \(\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le2\beta+1\), where \(\{\beta,\alpha\}\in\mathbb{R}\), are evaluated in terms of the hypergeometric functions \(_{1}F_2\) and \(_{2}F_3\), and their asymptotic expressions for \(|x|\gg1\) are also derived. The integrals of the form \(\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\) and \(\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\), where \(n\) is a positive integer, are expressed in terms \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\), and then evaluated. \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\) are also evaluated in terms of the hypergeometric function \(_{2}F_2\). And so, the hypergeometric functions, \(_{1}F_2\) and \(_{2}F_3\), are expressed in terms of \(_{2}F_2\). The exponential integral \(\text{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx\) where \(\beta\ge1\) and \(\alpha\le\beta+1\) and the logarithmic integral \(\text{Li}=\int_{\mu}^{x} dt/\ln{t}\), \(\mu>1\), are also expressed in terms of \(_{2}F_2\), and their asymptotic expressions are investigated. For instance, it is found that for \(x\gg2\), \(\text{Li}\sim {x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2-
\ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\), where the term \(\ln{\left({\ln{x}}/{\ln{2}}\right)}-2-
\ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\) is added to the known expression in mathematical literature \(\text{Li}\sim {x}/{\ln{x}}\). The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given. |
| format | Article |
| id | doaj-art-ecd61a2bb7ed4a3290b00f0ba4cd1569 |
| institution | Matheson Library |
| issn | 2414-3952 |
| language | English |
| publishDate | 2018-07-01 |
| 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-ecd61a2bb7ed4a3290b00f0ba4cd15692025-08-02T22:43:32ZengUral 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 Journal2414-39522018-07-014110.15826/umj.2018.1.00354EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART IVictor Nijimbere0School of Mathematics and Statistics, Carleton University, Ottawa, OntarioThe non-elementary integrals \(\text{Si}_{\beta,\alpha}=\int [\sin{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le\beta+1\) and \(\text{Ci}_{\beta,\alpha}=\int [\cos{(\lambda x^\beta)}/(\lambda x^\alpha)] dx,\) \(\beta\ge1,\) \(\alpha\le2\beta+1\), where \(\{\beta,\alpha\}\in\mathbb{R}\), are evaluated in terms of the hypergeometric functions \(_{1}F_2\) and \(_{2}F_3\), and their asymptotic expressions for \(|x|\gg1\) are also derived. The integrals of the form \(\int [\sin^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\) and \(\int [\cos^n{(\lambda x^\beta)}/(\lambda x^\alpha)] dx\), where \(n\) is a positive integer, are expressed in terms \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\), and then evaluated. \(\text{Si}_{\beta,\alpha}\) and \(\text{Ci}_{\beta,\alpha}\) are also evaluated in terms of the hypergeometric function \(_{2}F_2\). And so, the hypergeometric functions, \(_{1}F_2\) and \(_{2}F_3\), are expressed in terms of \(_{2}F_2\). The exponential integral \(\text{Ei}_{\beta,\alpha}=\int (e^{\lambda x^\beta}/x^\alpha) dx\) where \(\beta\ge1\) and \(\alpha\le\beta+1\) and the logarithmic integral \(\text{Li}=\int_{\mu}^{x} dt/\ln{t}\), \(\mu>1\), are also expressed in terms of \(_{2}F_2\), and their asymptotic expressions are investigated. For instance, it is found that for \(x\gg2\), \(\text{Li}\sim {x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\), where the term \(\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}\hspace{.075cm} _{2}F_{2}(1,1;2,2;\ln{2})\) is added to the known expression in mathematical literature \(\text{Li}\sim {x}/{\ln{x}}\). The method used in this paper consists of expanding the integrand as a Taylor and integrating the series term by term, and can be used to evaluate the other cases which are not considered here. This work is motivated by the applications of sine, cosine exponential and logarithmic integrals in Science and Engineering, and some applications are given.https://umjuran.ru/index.php/umj/article/view/108Non-elementary integralsSine integralCosine integralExponential integralLogarithmic integralHyperbolic sine integralHyperbolic cosine integralHypergeometric functionsAsymptotic evaluationFundamental theorem of calculus |
| spellingShingle | Victor Nijimbere EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I Ural Mathematical Journal Non-elementary integrals Sine integral Cosine integral Exponential integral Logarithmic integral Hyperbolic sine integral Hyperbolic cosine integral Hypergeometric functions Asymptotic evaluation Fundamental theorem of calculus |
| title | EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I |
| title_full | EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I |
| title_fullStr | EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I |
| title_full_unstemmed | EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I |
| title_short | EVALUATION OF SOME NON-ELEMENTARY INTEGRALS INVOLVING SINE, COSINE, EXPONENTIAL AND LOGARITHMIC INTEGRALS: PART I |
| title_sort | evaluation of some non elementary integrals involving sine cosine exponential and logarithmic integrals part i |
| topic | Non-elementary integrals Sine integral Cosine integral Exponential integral Logarithmic integral Hyperbolic sine integral Hyperbolic cosine integral Hypergeometric functions Asymptotic evaluation Fundamental theorem of calculus |
| url | https://umjuran.ru/index.php/umj/article/view/108 |
| work_keys_str_mv | AT victornijimbere evaluationofsomenonelementaryintegralsinvolvingsinecosineexponentialandlogarithmicintegralsparti |