On integrals of Bessel functions related to Weber's second exponential integral

Abstract

Two closely related integrals of Bessel functions are discussed: $$\int\limits_0^\infty {t^{1 - 2m} e^{ - {{t^2 } \mathord{\left/ {\vphantom {{t^2 } {2z}}} \right. \kern-\nulldelimiterspace} {2z}}} [J_n (t)]^2 } dt;\int\limits_0^z {u^{ - l} e^{ - u} I_n (u)} du.$$ Whenn, m, l are integers (m andl=1,2, ...,n), both integrals can be written as closed expressions in modified Bessel functionsI k (z). The results are interpreted in terms of hypergeometric seriesp F q. Series expansions inz and in 1/z are given.