Integrals and Asymptotic Expansions

Abstract

In assessing the content of Ramanujan’s first letter to him, Hardy [9, p. 9] judged that “on the whole, the integral formulae seemed the least imprsessive.” Later he added that Ramanujan’s definite integral formulae “are still interesting and will repay a careful analysis” [9, p. 186]. Indeed, a dismissal of Ramanujan’s contributions to intergration would have beed decidedly premature. First, we might recall that this first letter contained several remarkable formulas on series and continued fractions. In evaluating infinite series and deriving series identities, Ramanujan had no peers, except for possibly Euler and Jacobi. Ramanujan’s work on continued fraction expansions of analytic functions ranks as one of his most brilliant achievements. Thus, if Ramanujan’s contributions to integrals dim slightly in comparison, it is only because the glitter of diamonds surpasses that of rubies. Indeed, there are many elegant and important integrals that bear Ramanujan’s name (See for example, Entry 22.)