Inverse gamma function

In mathematics, the inverse gamma function is the inverse function of the gamma function. In other words, whenever. For example,. Usually, the inverse gamma function refers to the principal branch with domain on the real interval and image on the real interval, where is the minimum value of the gamma function on the positive real axis and is the location of that minimum.

Definition

The inverse gamma function may be defined by the following integral representation
where is a Borel measure such that and and are real numbers with.

To compute the branches of the inverse gamma function one can first compute the Taylor series of near. The series can then be truncated and inverted, which yields successively better approximations to. For instance, we have the quadratic approximation:
where is the trigamma function. The inverse gamma function also has the following asymptotic formula
where is the Lambert W function. The formula is found by inverting the Stirling approximation, and so can also be expanded into an asymptotic series.

To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the reciprocal gamma function near the poles at the negative integers, and then invert the series.
Setting then yields, for the n th branch of the inverse gamma function
where is the polygamma function.