Hadamard's gamma function


In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma 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:
where denotes the classical gamma function. If is a positive integer, then:

Properties

Unlike the classical gamma function, Hadamard's gamma function is an entire function, i.e. it has no poles in its domain. It satisfies the functional equation
with the understanding that is taken to be for positive integer values of.

Representations

Hadamard's gamma can also be expressed as
and as
where denotes the digamma function.