In mathematics, Ramanujan's master theorem (named after mathematician Srinivasa Ramanujan[1]) is a technique which provides an analytic expression for the Mellin transform of a function. Page from Ramanujan's notebook stating his Master theorem.
The result is stated as follows:
Assume function f(x) \! has an expansion of the form
In mathematics, Ramanujan's master theorem (named after mathematician Srinivasa Ramanujan[1]) is a technique which provides an analytic expression for the Mellin transform of a function.
ReplyDeletePage from Ramanujan's notebook stating his Master theorem.
The result is stated as follows:
Assume function f(x) \! has an expansion of the form
f(x)=\sum_{k=0}^\infty \frac{\phi(k)}{k!}(-x)^k \!
then Mellin transform of f(x) \! is given by
\int_0^\infty x^{s-1} f(x) \, dx = \Gamma(s)\phi(-s) \!
where \Gamma(s) \! is the Gamma function.
It was widely used by Ramanujan to calculate definite integrals and infinite series.
Multidimensional version of this theorem also appear in quantum physics (through Feynman diagrams).[2]
A similar result was also obtained by J. W. L. Glaisher.[http://en.wikipedia.org/wiki/Ramanujan%27s_master_theorem]