ON A FORMULA PROPOSED BY RAMANUJAN,S.

[摘要] This paper contains a detailed discussion concerning the validity of [GRAPHICS] which was proposed by S. Ramanujan in his second notebook. The formula, regarded either as a strict equality or as an asymptotic relation, is not valid for every continuous function phi(x) on R for which the integral is convergent. But it is shown that the formula holds asymptotically for phi(x) = a(x)f(x) whereby a represents a positive real parameter whose value can become as large as desired and f(x) can be 1, any positive integer power of x, any polynomial in x and any nonpolynomial function belonging to a wide class of functions each representable by its Maclaurin series expansion on R. Much attention is paid to the aspect of practical application of the obtained results. The article ends with the derivation of some remarkable identities following as a by-product from the calculations presented.

[发布日期] 1991-11-18

[关键词] ASYMPTOTIC FORMULAS OF RAMANUJAN