Ramanujan's master theorem

In mathematics, Ramanujan's master theorem, named after Srinivasa Ramanujan,^[1] is a technique that provides an analytic expression for the Mellin transform of an analytic function.

The result is stated as follows:

If a complex-valued function ${\textstyle f(x)}$ has an expansion of the form $${\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\,\varphi (k)\,}{k!}}(-x)^{k}}$$

then the Mellin transform of ${\textstyle f(x)}$ is given by

$${\displaystyle \int _{0}^{\infty }x^{s-1}f(x)\,dx=\Gamma (s)\,\varphi (-s)}$$

where ${\textstyle \Gamma (s)}$ is the gamma function.

It was widely used by Ramanujan to calculate definite integrals and infinite series.

Higher-dimensional versions of this theorem also appear in quantum physics through Feynman diagrams.^[2]

A similar result was also obtained by Glaisher.^[3]