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In [[mathematics]], '''Ramanujan's master theorem''' (named after mathematician [[Srinivasa Ramanujan]]<ref>B. Berndt. Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.</ref>) is a technique which provides an analytic expression for the [[Mellin transform]] of a function. | |||
[[Image:Ramanujanmasterth.jpg|thumb|Page from Ramanujan's notebook stating his Master theorem.]] | |||
The result is stated as follows: | |||
Assume function <math> f(x) \!</math> has an expansion of the form | |||
: <math> f(x)=\sum_{k=0}^\infty \frac{\phi(k)}{k!}(-x)^k \!</math> | |||
then [[Mellin transform]] of <math> f(x) \!</math> is given by | |||
: <math> \int_0^\infty x^{s-1} f(x) \, dx = \Gamma(s)\phi(-s) \!</math> | |||
where <math> \Gamma(s) \!</math> 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 diagram]]s).<ref>[http://129.81.170.14/~vhm/papers_html/RMT-GMS.pdf A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams by Iv´an Gonz´alez, V. H. Moll and Iv´an Schmidt]</ref> | |||
A similar result was also obtained by [[J. W. L. Glaisher]].<ref>J. W. L. Glaisher. A new formula in definite integrals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 48(315):53–55, Jul 1874.</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 original form after substituting <math> \lambda(n) = \frac{\phi(n)}{\Gamma(1+n)} \!</math> and using functional equation for [[Gamma function]]. | |||
The integral above is convergent for <math> 0< \operatorname{Re}(s)<1 \!</math>. | |||
== Proof == | |||
The proof of Ramanujan's Master Theorem provided by [[G. H. Hardy]]<ref>G. H. Hardy. Ramanujan. Twelve Lectures on subjects suggested by his life and work. [[Chelsea Publishing Company]], New York, N. Y., 3rd edition, 1978.</ref> employs [[Cauchy residue theorem|Cauchy's residue theorem]] as well as 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 [[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>. | |||
By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have following integral representation:<ref>O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 2. The Ramanujan Journal, 6:449–468, 2002.</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<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^{5/4}} \,dx = \sqrt{2} \frac{4\pi}{5} \zeta\left(\frac 5 4\right) </math> | |||
[[Mathematica|Mathematica 7]] is unable to compute these examples.<ref>Ramanujan's Master Theorem by Tewodros Amdeberhan, Ivan Gonzalez, Marshall Harrison, Victor H. Moll and Armin Straub, The Ramanujan Journal.</ref> | |||
== Evaluation of quartic integral == | |||
It is well known for the evaluation of | |||
: <math> F(a,m)=\int_0^\infty \frac{dx}{(x^4+2ax^2+1)^{m+1}} </math> | |||
which is a well known quartic integral.<ref>T. Amdeberhan and V. Moll. A formula for a quartic integral: a survey of old proofs and some new ones. The Ramanujan Journal, 18:91–102, 2009.</ref> | |||
== References == | |||
{{Reflist}} | |||
== External links == | |||
# http://mathworld.wolfram.com/RamanujansMasterTheorem.html | |||
# http://www.youtube.com/watch?v=gLp5OsfUlNE | |||
# http://arminstraub.com/files/publications/rmt.pdf | |||
[[Category:Srinivasa Ramanujan]] | |||
[[Category:Theorems in analytic number theory]] |
Revision as of 13:05, 16 January 2013
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.
The result is stated as follows:
Assume function has an expansion of the form
then Mellin transform of is given by
where 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.[3]
Alternative formalism
An alternative formulation of Ramanujan's master theorem is as follows:
which gets converted to original form after substituting and using functional equation for Gamma function.
The integral above is convergent for .
Proof
The proof of Ramanujan's Master Theorem provided by G. H. Hardy[4] employs Cauchy's residue theorem as well as the well-known Mellin inversion theorem.
Application to Bernoulli polynomials
The generating function of the Bernoulli polynomials is given by:
These polynomials are given in terms of Hurwitz zeta function:
by for . By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have following integral representation:[5]
Application to the Gamma function
Weierstrass's definition of the Gamma function
is equivalent to expression
where is the Riemann zeta function.
Then applying Ramanujan master theorem we have:
Mathematica 7 is unable to compute these examples.[6]
Evaluation of quartic integral
It is well known for the evaluation of
which is a well known quartic integral.[7]
References
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External links
- http://mathworld.wolfram.com/RamanujansMasterTheorem.html
- http://www.youtube.com/watch?v=gLp5OsfUlNE
- http://arminstraub.com/files/publications/rmt.pdf
- ↑ B. Berndt. Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.
- ↑ A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams by Iv´an Gonz´alez, V. H. Moll and Iv´an Schmidt
- ↑ J. W. L. Glaisher. A new formula in definite integrals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 48(315):53–55, Jul 1874.
- ↑ G. H. Hardy. Ramanujan. Twelve Lectures on subjects suggested by his life and work. Chelsea Publishing Company, New York, N. Y., 3rd edition, 1978.
- ↑ O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 2. The Ramanujan Journal, 6:449–468, 2002.
- ↑ Ramanujan's Master Theorem by Tewodros Amdeberhan, Ivan Gonzalez, Marshall Harrison, Victor H. Moll and Armin Straub, The Ramanujan Journal.
- ↑ T. Amdeberhan and V. Moll. A formula for a quartic integral: a survey of old proofs and some new ones. The Ramanujan Journal, 18:91–102, 2009.