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.

Page from Ramanujan's notebook stating his Master theorem.

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]

Alternative formalism

An alternative formulation of Ramanujan's Master Theorem is as follows:

${\displaystyle \int _{0}^{\infty }x^{s-1}\left(\,\lambda (0)-x\,\lambda (1)+x^{2}\,\lambda (2)-\,\cdots \,\right)dx={\frac {\pi }{\,\sin(\pi s)\,}}\,\lambda (-s)}$

which gets converted to the above form after substituting ${\textstyle \lambda (n)\equiv {\frac {\varphi (n)}{\,\Gamma (1+n)\,}}}$ and using the functional equation for the gamma function.

The integral above is convergent for ${\textstyle 0<\operatorname {\mathcal {Re}} (s)<1}$ subject to growth conditions on ${\textstyle \varphi }$. ^[4]

Proof

A proof subject to "natural" assumptions (though not the weakest necessary conditions) to Ramanujan's Master theorem was provided by G. H. Hardy ^[5] (chapter XI) employing the residue theorem and the well-known Mellin inversion theorem.

Application to Bernoulli polynomials

The generating function of the Bernoulli polynomials ${\textstyle B_{k}(x)}$ is given by:

${\displaystyle {\frac {z\,e^{x\,z}}{\,e^{z}-1\,}}=\sum _{k=0}^{\infty }B_{k}(x)\,{\frac {z^{k}}{k!}}}$

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

${\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{\,(n+a)^{s}\,}}}$

by ${\textstyle \zeta (1-n,a)=-{\frac {B_{n}(a)}{n}}}$ for ${\textstyle ~n\geq 1}$. Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation: ^[6]

${\displaystyle \int _{0}^{\infty }x^{s-1}\left({\frac {e^{-ax}}{\,1-e^{-x}\,}}-{\frac {1}{x}}\right)dx=\Gamma (s)\,\zeta (s,a)\!}$

which is valid for ${\textstyle 0<\operatorname {\mathcal {Re}} (s)<1}$.

Application to the gamma function

Weierstrass's definition of the gamma function

${\displaystyle \Gamma (x)={\frac {\,e^{-\gamma \,x\,}}{x}}\,\prod _{n=1}^{\infty }\left(\,1+{\frac {x}{n}}\,\right)^{-1}e^{x/n}\!}$

is equivalent to expression

${\displaystyle \log \Gamma (1+x)=-\gamma \,x+\sum _{k=2}^{\infty }{\frac {\,\zeta (k)\,}{k}}\,(-x)^{k}}$

where ${\textstyle \zeta (k)}$ is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

${\displaystyle \int _{0}^{\infty }x^{s-1}{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{2}}}\mathrm {d} x={\frac {\pi }{\sin(\pi s)}}{\frac {\zeta (2-s)}{2-s}}\!}$

valid for ${\textstyle 0<\operatorname {\mathcal {Re}} (s)<1}$.

Special cases of ${\textstyle s={\frac {1}{2}}}$ and ${\textstyle s={\frac {3}{4}}}$ are

${\displaystyle \int _{0}^{\infty }{\frac {\,\gamma x+\log \Gamma (1+x)\,}{x^{5/2}}}\,\mathrm {d} x={\frac {2\pi }{3}}\,\zeta \left({\frac {3}{2}}\right)}$

${\displaystyle \int _{0}^{\infty }{\frac {\,\gamma \,x+\log \Gamma (1+x)\,}{x^{9/4}}}\,\mathrm {d} x={\sqrt {2}}{\frac {4\pi }{5}}\zeta \left({\frac {5}{4}}\right)}$

Application to Bessel functions

The Bessel function of the first kind has the power series

${\displaystyle J_{\nu }(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{\Gamma (k+\nu +1)k!}}{\bigg (}{\frac {z}{2}}{\bigg )}^{2k+\nu }}$

By Ramanujan's Master Theorem, together with some identities for the gamma function and rearranging, we can evaluate the integral

${\displaystyle {\frac {2^{\nu -2s}\pi }{\sin {(\pi (s-\nu ))}}}\int _{0}^{\infty }z^{s-1-\nu /2}J_{\nu }({\sqrt {z}})\,dz=\Gamma (s)\Gamma (s-\nu )}$

valid for ${\textstyle 0<2\operatorname {\mathcal {Re}} (s)<\operatorname {\mathcal {Re}} (\nu )+{\tfrac {3}{2}}}$.

Equivalently, if the spherical Bessel function ${\textstyle j_{\nu }(z)}$ is preferred, the formula becomes

${\displaystyle {\frac {2^{\nu -2s}{\sqrt {\pi }}(1-2s+2\nu )}{\cos {(\pi (s-\nu ))}}}\int _{0}^{\infty }z^{s-1-\nu /2}j_{\nu }({\sqrt {z}})\,dz=\Gamma (s)\Gamma {\bigg (}{\frac {1}{2}}+s-\nu {\bigg )}}$

valid for ${\textstyle 0<2\operatorname {\mathcal {Re}} (s)<\operatorname {\mathcal {Re}} (\nu )+2}$.

The solution is remarkable in that it is able to interpolate across the major identities for the gamma function. In particular, the choice of ${\textstyle J_{0}({\sqrt {z}})}$ gives the square of the gamma function, ${\textstyle j_{0}({\sqrt {z}})}$ gives the duplication formula, ${\textstyle z^{-1/2}J_{1}({\sqrt {z}})}$ gives the reflection formula, and fixing to the evaluable ${\textstyle s={\frac {1}{2}}}$ or ${\textstyle s=1}$ gives the gamma function by itself, up to reflection and scaling.

Bracket integration method

The bracket integration method (method of brackets) applies Ramanujan's Master Theorem to a broad range of integrals. ^{[7] [8]} The bracket integration method generates an integral of a series expansion, introduces simplifying notations, solves linear equations, and completes the integration using formulas arising from Ramanujan's Master Theorem. ^[8]

Generate an integral of a series expansion

This method transforms the integral to an integral of a series expansion involving M variables, ${\textstyle x_{1},\ldots x_{M}}$, and S summation parameters, ${\textstyle n_{1},\dots n_{S}}$. A multivariate integral may assume this form. ^{[2] : 8}

${\displaystyle \int _{0}^{\infty }\cdots \int _{0}^{\infty }\sum _{n_{1},\ldots ,n_{S}=0}^{\infty }\varphi (n_{1}\cdots n_{S})\ \prod _{j=1}^{S}\left({\frac {(-1)^{n_{j}}}{n_{j}!}}\right)\prod _{j=1}^{M}(x_{j})^{(-c_{j}+a_{j1}\cdot n_{1}+\cdots +a_{jS}\cdot n_{S}-1)}\ dx_{1}\cdots dx_{M}}$

(B.0)

Apply special notations

• The bracket (${\textstyle \langle \cdots \rangle }$), indicator (${\textstyle \phi }$), and monomial power notations replace terms in the series expansion. ^{[2] : 8}

${\displaystyle \int _{0}^{\infty }x^{c+b\cdot n-1}\ dx\ \to \ <c+b\cdot n>}$

(B.1)

${\displaystyle {\frac {(-1)^{n}}{n!}}\ \to \ \phi _{n}}$

(B.2)

${\displaystyle \prod _{j=1}^{S}\left({\frac {(-1)^{n_{j}}}{n_{j}!}}\right)\ \to \ \phi _{n_{1},\ldots ,n_{S}}}$

(B.3)

${\displaystyle \left(\sum _{k=1}^{P}u_{k}\right)^{\mp d}\ \to \ \sum _{n_{1},\ldots ,n_{P}=0}^{\infty }\varphi _{n_{1},\ldots ,n_{P}}\prod _{k=1}^{P}u_{k}^{n_{k}}{\frac {\langle \pm d+\sum _{j=1}^{P}n_{j}\rangle }{\Gamma (\pm d)}}}$

(B.4)

• Application of these notations transforms the integral to a bracket series containing B brackets. ^{[7] : 56}

${\displaystyle \sum _{n_{1},\ldots ,n_{S}=0}^{\infty }\varphi (n_{1}\cdots n_{S})\ \phi (n_{1}\cdots n_{S})\prod _{j=1}^{B}\left\langle -c_{j}+\sum _{k=1}^{S}a_{jk}\cdot n_{k}\right\rangle }$

(B.5)

• Each bracket series has an index defined as index = number of sums − number of brackets.
• Among all bracket series representations of an integral, the representation with a minimal index is preferred. ^{[8] : 984}

Solve linear equations

• The array of coefficients ${\textstyle a_{jk}}$ must have maximum rank, linearly independent leading columns to solve the following set of linear equations. ^{[2] : 8  [8] : 985}
• If the index is non-negative, solve this equation set for each ${\textstyle n_{j}^{\ast }}$. The terms ${\textstyle n_{j}^{\ast }}$ may be linear functions of ${\textstyle \{n_{B+1}\dots n_{S}\}}$.

${\displaystyle -c_{j}+\sum _{k=1}^{B}a_{jk}\cdot n_{k}^{\ast }+\sum _{k=B+1}^{S}a_{jk}\cdot n_{k}=0}$

(B.6)

• If the index is zero, equation ( B.6 ) simplifies to solving this equation set for each ${\textstyle n_{j}^{\ast }}$

${\displaystyle -c_{j}+\sum _{k=1}^{B}a_{jk}\cdot n_{k}^{\ast }=0}$

(B.7)

• If the index is negative, the integral cannot be determined.

Apply formulas

• If the index is non-negative, the formula for the integral is this form. ^{[7] : 54}

${\displaystyle \sum _{n_{B+1}\dots n_{S}=0}^{\infty }{\frac {\varphi (n_{1}^{\ast }\dots n_{B}^{\ast },n_{B+1}\dots n_{S})\cdot \prod _{j=1}^{B}\Gamma (-n_{j}^{\ast })}{\det |A|}}}$

(B.8)

• These rules apply. ^{[8] : 985}
  • A series is generated for each choice of free summation parameters, ${\textstyle \{n_{B}+1,\dots N_{S}\}}$.
  • Series converging in a common region are added.
  • If a choice generates a divergent series or null series (a series with zero valued terms), the series is rejected.
  • A bracket series of negative index is assigned no value.
  • If all series are rejected, then the method cannot be applied.
  • If the index is zero, the formula B.8 simplifies to this formula and no sum occurs.

${\displaystyle \ {\frac {\varphi (n_{1}^{\ast }\dots n_{S}^{\ast })\cdot \prod _{j=1}^{S}\Gamma (-n_{j}^{\ast })}{\det |A|}}}$

(B.9)

Mathematical basis

• Apply this variable transformation to the general integral form ( B.0 ). ^{[4] : 14}

${\displaystyle y_{k}=x_{1}^{a_{1k}}\cdot \ldots \cdot x_{M}^{a_{Mk}}}$

(B.10)

.

• This is the transformed integral ( B.11 ) and the result from applying Ramanujan's Master Theorem ( B.12 ).

${\displaystyle (\det |A|^{-1})\cdot \int _{0}^{\infty }\cdots \int _{0}^{\infty }\sum _{n_{1},\ldots ,n_{S}=0}^{\infty }\varphi (n_{1}\cdots n_{S})\prod _{j=1}^{S}\left({\frac {(-1)^{n_{j}}}{n_{j}!}}\right)\prod _{j=1}^{M}(y_{j})^{-n_{j}^{\ast }+n_{j}-1}\ dy_{1}\cdots dy_{M}}$

(B.11)

${\displaystyle =\sum _{n_{M+1}\cdots n_{S}=0}^{\infty }{\frac {\varphi (n_{1}^{\ast }\dots n_{M}^{\ast },n_{M+1}\dots n_{S})\cdot \prod _{j=1}^{M}\Gamma (-n_{j}^{\ast })}{\det |A|}}}$

(B.12)

• The number of brackets (B) equals the number of integrals (M) ( B.1 ). In addition to generating the algorithm's formulas ( B.8 , B.9 ), the variable transformation also generates the algorithm's linear equations ( B.6 , B.7 ). ^{[4] : 14}

Example

• The bracket integration method is applied to this integral.

${\displaystyle \int _{0}^{\infty }x^{3/2}\cdot e^{-x^{3}/2}\ dx}$

• Generate the integral of a series expansion ( B.0 ).

${\displaystyle \int _{0}^{\infty }\sum _{n=0}^{\infty }2^{-n}\cdot {\frac {(-1)^{n}}{n!}}\cdot x^{(3\cdot n+5/2)-1}\ dx}$

• Apply special notations ( B.1 , B.2 ).

${\displaystyle \sum _{n=0}^{\infty }2^{-n}\cdot \phi (n)\cdot \langle 3\cdot n+{\frac {5}{2}}\rangle }$

• Solve the linear equation ( B.7 ).

${\displaystyle 3\cdot n^{\ast }+{\frac {5}{2}}=0,\ n^{\ast }={\frac {-5}{6}}}$

• Apply the formula ( B.9 ).

${\displaystyle {\frac {2^{5/6}\cdot \Gamma ({\frac {5}{6}})}{3}}}$

References

1. Berndt, B. (1985). Ramanujan's Notebooks, Part I. New York: Springer-Verlag.
2. González, Iván; Moll, V.H.; Schmidt, Iván (2011). "A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams". arXiv: 1103.0588 [math-ph].
3. Glaisher, J.W.L. (1874). "A new formula in definite integrals". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 48 (315): 53–55. doi:10.1080/14786447408641072.
4. Amdeberhan, Tewodros; Gonzalez, Ivan; Harrison, Marshall; Moll, Victor H.; Straub, Armin (2012). "Ramanujan's Master Theorem". The Ramanujan Journal. 29 (1–3): 103–120. CiteSeerX   10.1.1.232.8448 . doi:10.1007/s11139-011-9333-y. S2CID   8886049.
5. Hardy, G.H. (1978). Ramanujan: Twelve lectures on subjects suggested by his life and work (3rd ed.). New York, NY: Chelsea. ISBN   978-0-8284-0136-4.
6. Espinosa, O.; Moll, V. (2002). "On some definite integrals involving the Hurwitz zeta function. Part 2". The Ramanujan Journal. 6 (4): 449–468. arXiv: math/0107082 . doi:10.1023/A:1021171500736. S2CID   970603.
7. Gonzalez, Ivan; Moll, Victor H. (July 2010). "Definite integrals by the method of brackets. Part 1,". Advances in Applied Mathematics. 45 (1): 50–73. doi:10.1016/j.aam.2009.11.003.
8. Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020). "An extension of the method of brackets. Part 2". Open Mathematics. 18 (1): 983–995. arXiv: 1707.08942 . doi: 10.1515/math-2020-0062 . ISSN   2391-5455.

External links

• "Ramanujan's Master Theorem". mathworld.wolfram.com.
• "rmt" (PDF). ArminStraub. publications.