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Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan Integrals RS,C(m, n) in Terms of Hypergeometric Series 2F3(⋅)
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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.
Title: Analytical Expressions of Infinite Fourier Sine and Cosine Transform-Based Ramanujan Integrals RS,C(m, n) in Terms of Hypergeometric Series 2F3(⋅)
Description:
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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