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Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals. I. (English) Zbl 1455.33016

Summary: 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 \(_1F_2\) and \(_2F_3\), 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 \(_2F_2\). And so, the hypergeometric functions, \(_1F_2\) and \(_2F_3\), are expressed in terms of \(_2F_2\). 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 \(_2F_2\), and their asymptotic expressions are investigated. For instance, it is found that for \(x\gg2, \text{Li}\sim{x}/{\ln{x}}+\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}_2F_2(1,1;2,2;\ln{2})\), where the term \(\ln{\left({\ln{x}}/{\ln{2}}\right)}-2- \ln{2}_2F_2(1,1;2,2;\ln{2})\) 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.
For Part II, see [the author, ibid. 4, No. 1, 43–55 (2018; Zbl 1455.33017)].

MSC:

33F05 Numerical approximation and evaluation of special functions
33B10 Exponential and trigonometric functions
33C05 Classical hypergeometric functions, \({}_2F_1\)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Citations:

Zbl 1455.33017

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References:

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