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Hadamard's gamma function

Extension of the factorial function From Wikipedia, the free encyclopedia

Hadamard's gamma function
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In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance of a pseudogamma function). This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex numbers in a different way from Euler's gamma function. It is defined as:

Thumb
Hadamard's gamma function plotted over part of the real axis. Unlike the classical gamma function, it is holomorphic; there are no poles.

where Γ(x) denotes the classical gamma function. If n is a positive integer, then:

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Properties

Unlike the classical gamma function, Hadamard's gamma function H(x) is an entire function, i.e., it is defined and analytic at all complex numbers. It satisfies the functional equation

with the understanding that is taken to be 0 for positive integer values of x.

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Representations

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Hadamard's gamma can also be expressed as

and also as

where ψ(x) denotes the digamma function, and denotes the Lerch zeta function.

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See also

References

  • Hadamard, M. J. (1894), Sur L'Expression Du Produit 1·2·3· · · · ·(n−1) Par Une Fonction Entière (PDF) (in French), Œuvres de Jacques Hadamard, Centre National de la Recherche Scientifiques, Paris, 1968
  • Srivastava, H. M.; Junesang, Choi (2012). Zeta and Q-Zeta Functions and Associated Series and Integrals. Elsevier insights. p. 124. ISBN 978-0-12-385218-2.
  • "Introduction to the Gamma Function". The Wolfram Functions Site. Wolfram Research, Inc. Retrieved 27 February 2016.
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