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Add a bottom section.

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Hello, I would like to add a bottom section to enhance the Wikipedia page. The topic added is not large enough to merit a separate page. The additional content will enhance the quality of the Wikipedia page. The references overlap with the existing references. I added 2 additional references, added labels to the references, and capitalized Ramanujan's Master Theorem as it is a proper noun. Thanks.

Proposal

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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 has an expansion of the form

then the Mellin transform of is given by

where 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

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

which gets converted to the above form after substituting and using the functional equation for the gamma function.

The integral above is convergent for subject to growth conditions on .[4]

Proof

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

Application to Bernoulli polynomials

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The generating function of the Bernoulli polynomials is given by:

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

by for . Using the Ramanujan master theorem and the generating function of Bernoulli polynomials one has the following integral representation:[6]

which is valid for .

Application to the gamma function

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Weierstrass's definition of the gamma function

is equivalent to expression

where is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

valid for .

Special cases of and are

Application to Bessel functions

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The Bessel function of the first kind has the power series

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

valid for .

Equivalently, if the spherical Bessel function is preferred, the formula becomes

valid for .

The solution is remarkable in that it is able to interpolate across the major identities for the gamma function. In particular, the choice of gives the square of the gamma function, gives the duplication formula, gives the reflection formula, and fixing to the evaluable or gives the gamma function by itself, up to reflection and scaling.

Bracket Integration Method

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The Bracket Integration Method 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

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This method transforms the integral to an integral of a series expansion involving M variables, , and S summation parameters, . A multivariate integral may assume this form.[2]: 8 

(B.0)

Apply special notations

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  • The bracket (), indicator (), and monomial power notations replace terms in the series expansion.[2]: 8 
(B.1)
(B.2)
(B.3)
(B.4)
  • Application of these notations transforms the integral to a bracket series containing B brackets.[7]: 56 
(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

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  • The array of coefficients 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 . The terms may be linear functions of .
(B.6)
  • If the index is zero, equation (B.6) simplifies to solving this equation set for each
(B.7)
  • If the index is negative, the integral cannot be determined.


Apply formulas

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  • If the index is non-negative, the formula for the integral is this form.[7]: 54 
(B.8)
  • These rules apply.[8]: 985 
    • A series is generated for each choice of free summation parameters, .
    • 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.
(B.9)


Mathematical Basis

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  • Apply this variable transformation to the general integral form (B.0).[4]: 14 
(B.10)

.

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

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  • The Bracket Integration Method is applied to this integral.
  • Generate the integral of a series expansion (B.0).
  • Apply special notations (B.1, B.2).
  • Solve the linear equation (B.7).
  • Apply the formula (B.9).

References

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  1. ^ Berndt, B. (1985). Ramanujan's Notebooks, Part I. New York: Springer-Verlag.
  2. ^ a b c d 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. ^ a b c 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. ^ a b c 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. ^ a b c d e 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. doi:10.1515/math-2020-0062. ISSN 2391-5455.
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TMM53 (talk) 05:08, 7 October 2022 (UTC) TMM53 (talk) 05:08, 7 October 2022 (UTC)[reply]

Revision

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This is a proposed revision. Although current content is correct, a rewrite makes the content easier to understand for the reader. Additional content and references were added. If any concerns, please send me a message, and let us discuss. ThanksTMM53 (talk) 05:59, 21 May 2024 (UTC)[reply]