Английская Википедия:Hadamard's gamma function

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Файл:Hadamards gamma function plot.png

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, 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 than Euler's gamma function. It is defined as:

<math>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 \},</math>

where Шаблон:Math denotes the classical gamma function. If Шаблон:Math is a positive integer, then:

<math>H(n) = \Gamma(n) = (n-1)! </math>

Properties

Unlike the classical gamma function, Hadamard's gamma function Шаблон:Math is an entire function, i.e. it has no poles in its domain. It satisfies the functional equation

<math>H(x+1) = xH(x) + \frac{1}{\Gamma(1-x)},</math>

with the understanding that <math>\tfrac{1}{\Gamma(1-x)}</math> is taken to be Шаблон:Math for positive integer values of Шаблон:Mvar.

Representations

Hadamard's gamma can also be expressed as

<math>H(x)=\frac{\psi\left ( 1 - \frac{x}{2}\right )-\psi\left ( \frac{1}{2} - \frac{x}{2}\right )}{2\Gamma (1-x)} = \frac{\Phi\left(-1, 1, -x\right)}{\Gamma(-x)}</math>

where <math>\Phi</math> is the Lerch zeta function, and as

<math>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 ], </math>

where Шаблон:Math denotes the digamma function.

See also

• Gamma function
• Pseudogamma function

References

• Шаблон:Citation
• Шаблон:Cite book
• Шаблон:Cite web