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==== Approximation ====
To compute the branches of the inverse gamma function one can first compute the Taylor series of <math>\Gamma(x)</math> near <math>\alpha</math>. The series can then be truncated and inverted, which yields successively better approximations to <math>\Gamma^{-1}(x)</math>. For instance, we have the quadratic approximation <ref>{{cite journal |first1=Robert M.|last1=Corless |
<math>
\Gamma^{-1}\left(x\right)\approx\alpha+\sqrt{\frac{2\left(x-\Gamma\left(\alpha\right)\right)}{\Psi\left(1,\ \alpha\right)\Gamma\left(\alpha\right)}}.</math>
The inverse gamma function also has the following [[asymptotic formula]] <ref>{{cite journal |last1=Amenyou |first1=Folitse Komla |last2=Jeffrey |first2=David |title="Properties and Computation of the inverse of the Gamma Function" |date=2018 |pages=28
<math>\Gamma^{-1}(x)\sim\frac{1}{2}+\frac{\ln\left(\frac{x}{\sqrt{2\pi}}\right)}{W_{0}\left(e^{-1}\ln\left(\frac{x}{\sqrt{2\pi}}\right)\right)}</math>
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To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the [[reciprocal gamma function]] <math>\frac{1}{\Gamma(x)}</math> near the poles at the negative integers, and then invert the series.
Setting <math>z=\frac{1}{x}</math> then yields, for the ''n'' th branch <math>\Gamma_{n}^{-1}(z)</math> of the inverse gamma function (<math>n\ge 0</math>) <ref>{{Cite web |last=Couto |first=Ana Carolina Camargos |last2=Jeffrey |first2=David |last3=Corless |first3=Robert |date=November 2020 |title=The Inverse Gamma Function and its Numerical Evaluation |url=https://www.maplesoft.com/mapleconference/2020/highlights.aspx |url-status=live |place=Section 8 |
<math>\Gamma_{n}^{-1}(z)=-n+\frac{\left(-1\right)^{n}}{n!z}+\frac{\psi^{(0)}\left(n+1\right)}{\left(n!z\right)^2}+\frac{\left(-1\right)^{n}\left(\pi^{2}+9\psi^{(0)}\left(n+1\right)^{2}-3\psi^{(1)}\left(n+1\right)\right)}{6\left(n!z\right)^3}+O\left(\frac{1}{z^{4}}\right)</math>
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