Ramanujan's master theorem

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Short description: Mathematical 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 [math]\displaystyle{ f(x) }[/math] has an expansion of the form

[math]\displaystyle{ f(x)=\sum_{k=0}^\infty \frac{\,\varphi(k)\,}{k!}(-x)^k }[/math]

then the Mellin transform of [math]\displaystyle{ f(x) }[/math] is given by

[math]\displaystyle{ \int_0^\infty x^{s-1} f(x) \, dx = \Gamma(s)\,\varphi(-s) }[/math]

where [math]\displaystyle{ \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 diagrams).[2]

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

Alternative formalism

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

[math]\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) }[/math]

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

The integral above is convergent for [math]\displaystyle{ 0 \lt \operatorname{\mathcal{Re}}(s) \lt 1 }[/math] subject to growth conditions on [math]\displaystyle{ \varphi }[/math].[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 [math]\displaystyle{ B_k(x) }[/math] is given by:

[math]\displaystyle{ \frac{z\,e^{x\,z}}{\,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]\displaystyle{ \zeta(s,a) = \sum_{n=0}^\infty \frac{1}{\,(n+a)^s\,} }[/math]

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

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

which is valid for [math]\displaystyle{ 0 \lt \operatorname{\mathcal{Re}}(s) \lt 1 }[/math].

Application to the gamma function

Weierstrass's definition of the gamma function

[math]\displaystyle{ \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]\displaystyle{ \log\Gamma(1+x) = -\gamma\,x + \sum_{k=2}^\infty \frac{\,\zeta(k)\,}{k}\,(-x)^k }[/math]

where [math]\displaystyle{ \zeta(k) }[/math] is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

[math]\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} \! }[/math]

valid for [math]\displaystyle{ 0 \lt \operatorname{\mathcal{Re}}(s) \lt 1 }[/math].

Special cases of [math]\displaystyle{ s = \frac{1}{2} }[/math] and [math]\displaystyle{ s = \frac{3}{4} }[/math] are

[math]\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) }[/math]
[math]\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) }[/math]

Application to Bessel functions

The Bessel function of the first kind has the power series

[math]\displaystyle{ J_\nu(z)=\sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k+\nu+1)k!}\bigg(\frac{z}{2}\bigg)^{2k+\nu} }[/math]

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

[math]\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) }[/math]

valid for [math]\displaystyle{ 0 \lt 2\operatorname{\mathcal{Re}}(s) \lt \operatorname{\mathcal{Re}}(\nu)+\tfrac{3}{2} }[/math].

Equivalently, if the spherical Bessel function [math]\displaystyle{ j_\nu(z) }[/math] is preferred, the formula becomes

[math]\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) }[/math]

valid for [math]\displaystyle{ 0 \lt 2\operatorname{\mathcal{Re}}(s) \lt \operatorname{\mathcal{Re}}(\nu)+2 }[/math].

The solution is remarkable in that it is able to interpolate across the major identities for the gamma function. In particular, the choice of [math]\displaystyle{ J_0(\sqrt z) }[/math] gives the square of the gamma function, [math]\displaystyle{ j_0(\sqrt{z}) }[/math] gives the duplication formula, [math]\displaystyle{ z^{-1/2}J_{1}(\sqrt z) }[/math] gives the reflection formula, and fixing to the evaluable [math]\displaystyle{ s=\frac{1}{2} }[/math] or [math]\displaystyle{ s=1 }[/math] 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, [math]\displaystyle{ x_1, \ldots x_M }[/math], and S summation parameters, [math]\displaystyle{ n_1, \dots n_S }[/math]. A multivariate integral may assume this form.[2]:8

[math]\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 }[/math]

 

 

 

 

(B.0)

Apply special notations

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

[math]\displaystyle{ \int_0^\infty x^{c+b \cdot n -1} \ dx \ \to \ \lt c+b \cdot n \gt }[/math]

 

 

 

 

(B.1)

[math]\displaystyle{ \frac{(-1)^n}{n !} \ \to \ \phi_n }[/math]

 

 

 

 

(B.2)

[math]\displaystyle{ \prod_{j=1}^S \left( \frac{(-1)^{n_j}}{n_j!} \right) \ \to \ \phi_{n_1,\ldots,n_S} }[/math]

 

 

 

 

(B.3)

[math]\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)} }[/math]

 

 

 

 

(B.4)

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

[math]\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 }[/math]

 

 

 

 

(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 [math]\displaystyle{ a_{jk} }[/math] 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 [math]\displaystyle{ n^{\ast}_{j} }[/math]. The terms [math]\displaystyle{ n^{\ast}_{j} }[/math] may be linear functions of [math]\displaystyle{ \{n_{B+1} \dots n_{S} \} }[/math].

[math]\displaystyle{ -c_j+\sum_{k=1}^B a_{jk} \cdot n^\ast_k + \sum_{k=B+1}^S a_{jk} \cdot n_k = 0 }[/math]

 

 

 

 

(B.6)

  • If the index is zero, equation (B.6) simplifies to solving this equation set for each [math]\displaystyle{ n^\ast_j }[/math]

[math]\displaystyle{ -c_j+\sum_{k=1}^B a_{jk} \cdot n^\ast_k=0 }[/math]

 

 

 

 

(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

[math]\displaystyle{ \sum_{n_{B+1} \dots n_S=0}^\infty \frac{\varphi(n^\ast_1 \dots n^\ast_B, n_{B+1} \dots n_S) \cdot \prod_{j=1}^B \Gamma (- n^\ast_j)}{\det|A|} }[/math]

 

 

 

 

(B.8)

  • These rules apply.[8]:985
    • A series is generated for each choice of free summation parameters, [math]\displaystyle{ \{ n_B+1, \dots N_S \} }[/math].
    • 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.

[math]\displaystyle{ \ \frac{\varphi(n^\ast_1 \dots n^\ast_S) \cdot \prod_{j=1}^S \Gamma (- n^\ast_j)}{\det|A|} }[/math]

 

 

 

 

(B.9)

Mathematical basis

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

[math]\displaystyle{ y_k= x_1^{a_{1k}} \cdot \ldots \cdot x_{M}^{a_{Mk}} }[/math]

 

 

 

 

(B.10)

.

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

[math]\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^\ast_j +n_j -1 } \ dy_1 \cdots dy_M }[/math]

 

 

 

 

(B.11)

[math]\displaystyle{ = \sum_{n_{M+1} \cdots n_S=0}^\infty \frac{\varphi(n^{\ast}_{1} \dots n^\ast_M, n_{M+1} \dots n_S) \cdot \prod_{j=1}^M \Gamma (- n^\ast_j)}{\det|A|} }[/math]

 

 

 

 

(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.
[math]\displaystyle{ \int_0^\infty x^{3/2} \cdot e^{-x^3/2} \ dx }[/math]
  • Generate the integral of a series expansion (B.0).
[math]\displaystyle{ \int_0^\infty \sum_{n=0}^\infty 2^{-n} \cdot \frac{(-1)^n}{n!} \cdot x^{(3 \cdot n+5/2)-1} \ dx }[/math]
  • Apply special notations (B.1, B.2).
[math]\displaystyle{ \sum_{n=0}^\infty 2^{-n} \cdot \phi(n) \cdot \langle 3 \cdot n+ \frac{5}{2} \rangle }[/math]
  • Solve the linear equation (B.7).
[math]\displaystyle{ 3 \cdot n^\ast + \frac{5}{2}=0, \ n^\ast= \frac{-5}{6} }[/math]
  • Apply the formula (B.9).
[math]\displaystyle{ \frac {2^{5/6} \cdot \Gamma(\frac{5}{6})}{3} }[/math]

References

  1. Berndt, B. (1985). Ramanujan's Notebooks, Part I. New York: Springer-Verlag. 
  2. 2.0 2.1 2.2 2.3 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. 4.0 4.1 4.2 Amdeberhan, Tewodros; Gonzalez, Ivan; Harrison, Marshall; Moll, Victor H.; Straub, Armin (2012). "Ramanujan's Master Theorem". The Ramanujan Journal 29 (1–3): 103–120. doi:10.1007/s11139-011-9333-y. 
  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. doi:10.1023/A:1021171500736. 
  7. 7.0 7.1 7.2 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. 8.0 8.1 8.2 8.3 8.4 Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020). "An extension of the method of brackets. Part 2" (in en). Open Mathematics 18 (1): 983–995. doi:10.1515/math-2020-0062. ISSN 2391-5455. https://www.degruyter.com/document/doi/10.1515/math-2020-0062/html?lang=en. 

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