Difference between revisions of "Hadamard gamma"
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| + | 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= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> 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]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
[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?] | ||
Revision as of 22:52, 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:
Proof: proof goes here █
