Reciprocal gamma function

In mathematics, the reciprocal gamma function is the function
where 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, but of infinite type.
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.

Infinite product expansion

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 is the Euler–Mascheroni constant. These expansions are valid for all complex numbers .

Taylor series

expansion around gives:
where is the Euler–Mascheroni constant. For, the coefficient for the term can be computed recursively as
where is the Riemann zeta function. An integral representation for these coefficients was recently found by Fekih-Ahmed :
For small values, these give the following values:
Fekih-Ahmed also gives an approximation for :
where, and is the negative-first branch of the Lambert function.
The Taylor expansion around has the same coefficients, i.e.:
.

Asymptotic expansion

As goes to infinity at a constant we have:

Contour integral representation

An integral representation due to Hermann Hankel is
where 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. According to Schmelzer & Trefethen, numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function.

Integral representations at the positive integers

For positive integers, there is an integral for the reciprocal factorial function given by
Similarly, for any real and such that we have the next integral for the reciprocal gamma function along the real axis in the form of:
where the particular case when provides a corresponding relation for the reciprocal double factorial function,

Integral along the real axis

Integration of the reciprocal gamma function along the positive real axis gives the value
which is known as the Fransén-Robinson constant.
We have the following formula