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Add a bottom section.
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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.
A similar result was also obtained by Glaisher.[3]
Alternative formalism
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
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.
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
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
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
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
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
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
The Bracket Integration Method is applied to this integral.
Generate the integral of a series expansion (B.0).
^Berndt, B. (1985). Ramanujan's Notebooks, Part I. New York: Springer-Verlag.
^ abcdGonzález, Iván; Moll, V.H.; Schmidt, Iván (2011). "A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams". arXiv:1103.0588 [math-ph].
^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.
^ abcGonzalez, 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.
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]