Hadamard's gamma function

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

In mathematics, the Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the gamma 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 to Euler's Gamma function. It is defined as:

H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma (1-{\frac {x}{2}})}}\right)\right\}

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

H(n)=\Gamma (n)=(n-1)!\,

Properties

Unlike the classical Gamma function, Hadamard's gamma function $H (x)$ is an entire function, i.e. it has no poles in its domain. It satisfies the functional equation

H(x+1)=x\,H(x)+{\frac {1}{\Gamma (1-x)}}

Note that ${\tfrac {1}{\Gamma (1-x)}}\to 0$ for the positive integer values of $x$ .

Representations

Hadamard's gamma can be expressed in terms of digamma functions as

H(x)={\frac {\psi \left(1-{\frac {x}{2}}\right)-\psi \left({\frac {1}{2}}-{\frac {x}{2}}\right)}{2\,\Gamma (1-x)}}

and as

H(x)=\Gamma (x)\left[1+{\frac {\sin(\pi x)}{2\pi }}\left\{\psi \left({\dfrac {x}{2}}\right)-\psi \left({\dfrac {x+1}{2}}\right)\right\}\right],

where $ψ(x)$ denotes the digamma function.

References

Hadamard, M. J. (1894), Sur L’Expression Du Produit 1·2·3· · · · ·(n−1) Par Une Fonction Entière (PDF) (in French), OEuvres 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 0123852188.
"Introduction to the Gamma Function". The Wolfram Functions Site. Wolfram Research, Inc. Retrieved 27 February 2016.

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