Inverse gamma function: Difference between revisions

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Setting <math>z=\frac{1}{x}</math> then yields, for the nth branch 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 |publication-place=Maple Conference Proceedings}}</ref>:

<math>\Gamma^{-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(x\frac{1}{z^{4}}\right)</math>

Where <math>\psi^{(n)}(x)</math> is the [[polygamma function]].