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Concept
Hadamard's gamma function
Summary
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:
where Γ(x) denotes the classical gamma function. If n is a positive integer, then:
Unlike the classical gamma function, Hadamard's gamma function H(x) 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 0 for positive integer values of x.
Hadamard's gamma can also be expressed as
and as
where ψ(x) denotes the digamma function.
Official source
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