View source for Ramanujan's master theorem

In [[mathematics]], '''Ramanujan's master theorem''' (named after [[Srinivasa Ramanujan]]<ref>{{cite book |first=B. |last=Berndt |title=Ramanujan's Notebooks, Part I |publisher=Springer-Verlag |location=New York |year=1985 }}</ref>) is a technique that provides an analytic expression for the [[Mellin transform]] of an [[analytic function]].
[[File:Ramanujan's "Master Theorem" page.jpg|thumb|Page from Ramanujan's notebook stating his Master theorem.]]

The result is stated as follows:

If a complex-valued function <math> f(x) </math> has an expansion of the form

: <math> f(x)=\sum_{k=0}^\infty \frac{\varphi(k)}{k!}(-x)^k \!</math>

then the [[Mellin transform]] of <math> f(x) </math> is given by

: <math> \int_0^\infty x^{s-1} f(x) \, dx = \Gamma(s)\varphi(-s) \!</math>

where <math> \Gamma(s) \!</math> 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 diagram]]s).<ref>{{cite arXiv |title=A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams |first=Iván |last=González |first2=V. H. |last2=Moll |first3=Iván |last3=Schmidt |eprint = 1103.0588|class = math-ph|year = 2011}}</ref>

A similar result was also obtained by [[J. W. L. Glaisher]].<ref>{{cite journal |first=J. W. L. |last=Glaisher |title=A new formula in definite integrals |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=48 |issue=315 |pages=53–55 |year=1874 |doi=10.1080/14786447408641072 }}</ref>

== Alternative formalism ==
An alternative formulation of Ramanujan's master theorem is as follows:

: <math> \int_0^\infty  x^{s-1} ({\lambda(0)-x\lambda(1)+x^{2}\lambda(2)-\cdots}) \, dx = \frac{\pi}{\sin(\pi s)}\lambda(-s) </math>

which gets converted to the above form after substituting <math> \lambda(n) = \frac{\varphi(n)}{\Gamma(1+n)} </math> and using the functional equation for the [[gamma function]].

The integral above is convergent for <math> 0< \operatorname{Re}(s)<1 </math> subject to growth conditions on <math> \varphi </math>.<ref>{{cite journal |title=Ramanujan's Master Theorem |first=Tewodros |last=Amdeberhan |first2=Ivan |last2=Gonzalez |first3=Marshall |last3=Harrison |first4=Victor H. |last4=Moll |first5=Armin |last5=Straub |journal=The Ramanujan Journal |volume=29 |issue=1–3 |year=2012 |pages=103–120 |doi=10.1007/s11139-011-9333-y |citeseerx=10.1.1.232.8448 }}</ref>

== Proof ==
A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by [[G. H. Hardy]]<ref>{{cite book |first=G. H. |last=Hardy |title=Ramanujan. Twelve Lectures on subjects suggested by his life and work |publisher=Chelsea |location=New York |edition=3rd |year=1978 |isbn=978-0-8284-0136-4 }}</ref> employing the [[residue theorem]] and the well-known [[Mellin inversion theorem]].

== Application to Bernoulli polynomials ==
The generating function of the [[Bernoulli polynomials]] <math> B_k(x)</math> is given by:

: <math> \frac{ze^{xz}}{e^z-1}=\sum_{k=0}^\infty  B_k(x)\frac{z^k}{k!} \!</math>

These polynomials are given in terms of the [[Hurwitz zeta function]]:

: <math> \zeta(s,a)=\sum_{n=0}^\infty  \frac{1}{(n+a)^s} \!</math>

by <math> \zeta(1-n,a)=-\frac{B_n(a)}{n} \!</math> for <math> n\geq1 </math>.
Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:<ref>{{cite journal |first=O. |last=Espinosa |first2=V. |last2=Moll |title=On some definite integrals involving the Hurwitz zeta function. Part 2 |journal=The Ramanujan Journal |volume=6 |issue=4 |pages=449–468 |year=2002 |doi=10.1023/A:1021171500736 |arxiv=math/0107082 }}</ref>

: <math> \int_0^\infty  x^{s-1} \left(\frac{e^{-ax}}{1-e^{-x}}-\frac{1}{x}\right) \, dx = \Gamma(s)\zeta(s,a) \!</math>

valid for <math> 0<\operatorname{Re}(s)<1\!</math>.

== Application to the Gamma function ==
Weierstrass's definition of the Gamma function

: <math> \Gamma(x)=\frac{e^{-\gamma x}}{x}\prod_{n=1}^\infty \left(1+\frac{x}{n}\right)^{-1} e^{x/n} \!</math>

is equivalent to expression

: <math> \log\Gamma(1+x)=-\gamma x+\sum_{k=2}^\infty \frac{\zeta(k)}{k}(-x)^k \!</math>

where <math> \zeta(k) </math> is the [[Riemann zeta function]].

Then applying Ramanujan master theorem we have:

: <math> \int_0^\infty x^{s-1} \frac{\gamma x+\log\Gamma(1+x)}{x^2} \, dx= \frac{\pi}{\sin(\pi s)}\frac{\zeta(2-s)}{2-s} \!</math>

valid for <math> 0<Re(s)<1\!</math>.

Special cases of <math> s=\frac{1}{2} </math> and <math> s=\frac{3}{4} </math> are

: <math> \int_0^\infty \frac{\gamma x+\log\Gamma(1+x)}{x^{5/2}} \, dx =\frac{2\pi}{3} \zeta\left( \frac{3}{2} \right) </math>

: <math> \int_0^\infty \frac{\gamma x+\log\Gamma(1+x)}{x^{9/4}} \,dx = \sqrt{2} \frac{4\pi}{5} \zeta\left(\frac 5 4\right) </math>

== References ==
{{Reflist}}

== External links ==
* http://mathworld.wolfram.com/RamanujansMasterTheorem.html
* http://arminstraub.com/files/publications/rmt.pdf

[[Category:Srinivasa Ramanujan]]
[[Category:Theorems in analytic number theory]]