On a formula proposed by S. Ramanujan (Q1184772)

This paper contains interesting results concerning the validity of \[ \int_ 0^{+\infty}\phi(x)/x^ x dx=\sum_{k=- \infty}^{+\infty}\phi(k)/k^ k, \] which was proposed by Ramanujan in his second notebook. It is shown that the formula, regarded either as a strict equality or as an asymptotic relation, is not valid for every continuous function \(\phi(x)\) for which the integral is convergent. Furthermore, the author proves that the formula holds asymptotically for \(phi(x)=a^ x f(x)\), where \(a\) is a positive real parameter and \(f(x)\) belongs to a certain wide class of functions.