Difference between revisions of "Hadamard gamma"
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| − | [http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html Is the Gamma function misdefined?] | + | [http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html Is the Gamma function misdefined?]<br /> |
[http://www.jstor.org/discover/10.2307/2309786?sid=21105065140641&uid=3739256&uid=2129&uid=70&uid=3739744&uid=4&uid=2 Leonhard Euler's Integral: A Historical Profile of the Gamma Function] | [http://www.jstor.org/discover/10.2307/2309786?sid=21105065140641&uid=3739256&uid=2129&uid=70&uid=3739744&uid=4&uid=2 Leonhard Euler's Integral: A Historical Profile of the Gamma Function] | ||
Revision as of 23:01, 13 January 2015
The Hadamard gamma function is defined by the formula $$H(x)=\dfrac{1}{\Gamma(1-x)} \dfrac{d}{dx} \log \left( \dfrac{\Gamma(\frac{1}{2}-\frac{x}{2})}{\Gamma(1-\frac{x}{2})} \right),$$ where $\Gamma$ denotes the gamma function.
Properties
Theorem: We can write $$H(x)=\dfrac{\psi(1-\frac{x}{2})-\psi(\frac{1}{2}-\frac{x}{2})}{2\Gamma(1-x)},$$ where $\psi$ is the digamma function.
Proof: proof goes here █
Theorem: The function $H$ is an entire function.
Proof: proof goes here █
Theorem: The function $H$ satisfies the formula $$H(x+1)=xH(x)+\dfrac{1}{\Gamma(1-x)}.$$
Proof: proof goes here █
References
Is the Gamma function misdefined?
Leonhard Euler's Integral: A Historical Profile of the Gamma Function
