Prove Ramanujan's theorem (in detail)in MSc.
Answers
Answer:
The result is stated as follows:
If a complex-valued function {\displaystyle f(x)}f(x) has an expansion of the form
{\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\varphi (k)}{k!}}(-x)^{k}\!}{\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\varphi (k)}{k!}}(-x)^{k}\!}
then the Mellin transform of {\displaystyle f(x)}f(x) is given by
{\displaystyle \int _{0}^{\infty }x^{s-1}f(x)\,dx=\Gamma (s)\varphi (-s)\!}{\displaystyle \int _{0}^{\infty }x^{s-1}f(x)\,dx=\Gamma (s)\varphi (-s)\!}
where {\displaystyle \Gamma (s)\!}\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 J. W. L. Glaisher.[3]
