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Concept
Reciprocal gamma function
Résumé
In mathematics, the reciprocal gamma function is the function
where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that log log grows no faster than log ), but of infinite type (meaning that log grows faster than any multiple of , since its growth is approximately proportional to log in the left-half plane).
The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function.
Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.
Following from the infinite product definitions for the gamma function, due to Euler and Weierstrass respectively, we get the following infinite product expansion for the reciprocal gamma function:
where γ = 0.577216... is the Euler–Mascheroni constant. These expansions are valid for all complex numbers z.
Taylor series expansion around 0 gives:
where γ is the Euler–Mascheroni constant. For n > 2, the coefficient an for the zn term can be computed recursively as
where ζ is the Riemann zeta function. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):
For small values, these give the following values:
Fekih-Ahmed (2014) also gives an approximation for :
where and is the minus-first branch of the Lambert W function.
The Taylor expansion around 1 has the same (but shifted) coefficients, i.e.:
(the reciprocal of Gauss' pi-function).
As goes to infinity at a constant arg(z) we have:
An integral representation due to Hermann Hankel is
where H is the Hankel contour, that is, the path encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis.
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