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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 grows no faster than ), but of infinite type (meaning that grows faster than any multiple of |z|, since its growth is approximately proportional to in the left-hand 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.
where C is a 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 along the real axis
Integration of the reciprocal gamma function along the positive real axis gives the value