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 .
Series Expansions Let be the -th branch of the gamma function, with denoting the principal branch. To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the reciprocal gamma function near the zeros at the negative integers, and then invert the series.
Setting in the expansion then yields, for :where is the polygamma function.
This can be rigorously justified by the Lagrange inversion theorem, which says that if where is analytic at a point and , then the inverse is given by a power series
For example, for the first non principal branch, set and such that . The formula gives
Now, assuming is invertible when restricted to an appropriate region, setting gives
Approximation 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.
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