Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan Integrals <em>R</em><sub><em>S,C</em></sub>(<em>m, n</em>) in Terms of Hypergeometric Series <sub>2</sub><em>F</em><sub>3</sub>(⋅)
In this chapter, we obtain analytical expressions of infinite Fourier sine and cosine transform-based Ramanujan integrals, RS,Cmn=∫0∞xm−1+exp2πxsincosπnxdx, in an infinite series of hypergeometric functions 2F3⋅, using the hypergeometric technique. Also, we have given some generalizations of the Ram...
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| Main Authors: | , |
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| Format: | Book Chapter |
| Language: | English |
| Published: |
IntechOpen
2023
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| Subjects: | |
| Online Access: | https://www.intechopen.com/chapters/84869 |
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| Summary: | In this chapter, we obtain analytical expressions of infinite Fourier sine and cosine transform-based Ramanujan integrals, RS,Cmn=∫0∞xm−1+exp2πxsincosπnxdx, in an infinite series of hypergeometric functions 2F3⋅, using the hypergeometric technique. Also, we have given some generalizations of the Ramanujan’s integrals RS,Cmn in the form of integrals denoted by IS,C∗υbcλy,JS,Cυbcλy,KS,Cυbcλy and IS,Cυbλy. These generalized definite integrals are expressed in terms of ordinary hypergeometric functions 2F3⋅, with suitable convergence conditions. Moreover, as applications of Ramanujan’s integrals RS,Cmn, some closed form of infinite summation formulas involving hypergeometric functions 1F2, 2F3⋅, and 0F1 are derived. |
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| ISBN: | 978-1-83768-459-5 |
| DOI: | 10.5772/intechopen.108401 |