On An Interesting Integral Involving Gauss’s Hypergeometric Function

Shoukat Ali

Abstract


Around five decades ago, MacRobert, in his very popular, useful and interesting research paper, obtained certain new class of finite integrals. He, then used the integrals to evaluate integrals involving E-functions. The aim of this short research paper is to find an interesting integral involving hypergeometric function by employing one of the integrals obtained by MacRobert. The beauty of our result is that it comes out in terms of products of the ratios of gamma function. Also, we know that whenever the value of an integral comes in terms of gamma functions, the results are very useful from the applications point of view. On specializing the parameters, we can easily obtain the known integrals due to Rathie and integral given in the book of Mathai and Saxena. It is not out of place to mention here that, for other integrals, transformation and summation formulas involving hypergeoemtric and generalized hypergeometric series, we refer the work by Ali. Applications of this integral in obtaining integrals involving Inayat Hussain’s -function, Rathie's -function etc. are under investigations and will be published soon.


Keywords


Hypergeometric functions; Integrals

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