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 . 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:

 

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. Series expansion 
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. References 

Gamma and related functions