2016/05/28 by Dixit, Atul, Roy, Arindam, Zaharescu, Alexandru · 1 citation
#33B20 #34E05 #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 33C15
paper · doi:10.48550/arxiv.1605.08904
A new transformation involving the error function \textuperf(z), the imaginary error function \textuperfi(z), and an integral analogue of a partial theta function is given along with its character analogues. Another complementary error function transformation is also obtained which when combined with the first explains a transformation in Ramanujan's Lost Notebook termed by Berndt and Xu as the one for an integral analogue of theta functions. These transformations are used to obtain a variety of exact and approximate evaluations of some non-elementary integrals involving hypergeometric functions. Several asymptotic expansions, including the one for a non-elementary integral involving a product of the Riemann Ξ-function of two different arguments, are obtained, which generalize known results due to Berndt and Evans, and Oloa.