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, 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:

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

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

$H(n)=\mathrm{\Gamma }(n)=(n-1)!$

Properties

Unlike the classical gamma function, Hadamard's gamma function Template:Math is an entire function, i.e., it is defined and analytic at all complex numbers. It satisfies the functional equation

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

with the understanding that $\frac{1}{\mathrm{\Gamma }(1-x)}$ is taken to be Template:Math for positive integer values of Template:Mvar.

The Hadamard's gamma function has a superadditive property:

$H(x)+H(y)\le H(x+y),$

for all $x,y\ge \alpha$, where $\alpha =\mathrm{1.5031...}$ Template:OEIS is the unique solution to the equation $H(2t)=2H(t)$ in the interval $[1.5,\mathrm{\infty })$.

Representations

Hadamard's gamma can also be expressed as

$H(x)=\frac{\psi (1-\frac{x}{2})-\psi (\frac{1}{2}-\frac{x}{2})}{2\mathrm{\Gamma }(1-x)}=\frac{L(-1,1,-x)}{\mathrm{\Gamma }(-x)},$

and also as

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

where Template:Math denotes the digamma function, and $L$ denotes the Lerch zeta function.

See also

• Gamma function
• Pseudogamma function

References

• Template:Citation
• Template:Cite book
• Template:Cite web