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Hadamard's gamma function#Properties

{{Short description|Extension of the factorial function}}

File:Hadamards gamma function plot.png

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:

: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 {{math|Γ(x)}} denotes the classical gamma function. If {{math|n}} is a positive integer, then:

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

Properties

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

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

with the understanding that \tfrac{1}{\Gamma(1-x)} is taken to be {{math|0}} for positive integer values of {{mvar|x}}.

Representations

Hadamard's gamma can also be expressed as

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

and also 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 {{math|ψ(x)}} denotes the digamma function, and L denotes the Lerch zeta function.

See also

References

  • {{citation |first=M. J. |last=Hadamard

|title=Sur L'Expression Du Produit 1·2·3· · · · ·(n−1) Par Une Fonction Entière

|publisher=Œuvres de Jacques Hadamard, Centre National de la Recherche Scientifiques, Paris, 1968

|url=http://www.luschny.de/math/factorial/hadamard/HadamardFactorial.pdf|year=1894 |language=fr}}

  • {{cite book |last1=Srivastava |first1=H. M. |last2=Junesang |first2=Choi |title=Zeta and Q-Zeta Functions and Associated Series and Integrals |publisher=Elsevier insights |year=2012 |isbn=978-0-12-385218-2 |pages=124}}
  • {{cite web |website=The Wolfram Functions Site |url=http://functions.wolfram.com/GammaBetaErf/Gamma/introductions/Gamma/ShowAll.html |title=Introduction to the Gamma Function |publisher=Wolfram Research, Inc |access-date=27 February 2016}}

Category:Gamma and related functions

Category:Analytic functions

Category:Special functions