Inverse gamma function explained

In mathematics, the inverse gamma function

\Gamma ^{-1} (x) is the inverse function of the gamma function. In other words, y = \Gamma ^{-1} (x) whenever $\backslash Gamma(y)=x$. For example, \Gamma ^{-1} ( 24 ) =5 .^[1] Usually, the inverse gamma function refers to the principal branch with domain on the real interval \left[\beta, +inf ty\right) and image on the real interval \left[\alpha, +inf ty\right) , where \beta = 0 . 8856031 \ldots is the minimum value of the gamma function on the positive real axis and \alpha = \Gamma ^{-1} (\beta) = 1 . 4616321 \ldots is the location of that minimum.^[2]

Definition

The inverse gamma function may be defined by the following integral representation^[3] $$\backslash Gamma^(x)=a+bx+\backslash int\_^\backslash left(\backslash frac-\backslash frac\backslash right)d\backslash mu(t)\backslash ,,$$where

\mu (t) is a Borel measure such that and a and b are real numbers with b \geqq 0 .

Approximation

To compute the branches of the inverse gamma function one can first compute the Taylor series of

\Gamma(x) near \alpha . The series can then be truncated and inverted, which yields successively better approximations to \Gamma ^{-1} (x) . For instance, we have the quadratic approximation:^[4]

$$\backslash Gamma^\backslash left(x\backslash right)\backslash approx\backslash alpha+\backslash sqrt.$$

The inverse gamma function also has the following asymptotic formula^[5] $$\backslash Gamma^(x)\backslash sim\backslash frac+\backslash frac\backslash ,,$$where

W _{0(x)} 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

(1)/(\Gamma(x)) near the poles at the negative integers, and then invert the series.

Setting

z = (1)/(x) then yields, for the n th branch \Gamma ^{-1} _{n} (z) of the inverse gamma function (n\ge 0 )^[6] $$\backslash Gamma\_^(z)=-n+\backslash frac+\backslash frac+\backslash frac+O\backslash left(\backslash frac\backslash right)\backslash ,,$$where \psi ^{(n)} (x) is the polygamma function.

Notes and References

1. Borwein . Jonathan M. . Corless . Robert M.. Gamma and Factorial in the Monthly . The American Mathematical Monthly . 2017 . 125 . 5 . 400–424 . 10.1080/00029890.2018.1420983 . 1703.05349 . 48663320 . 119324101.
2. Uchiyama . Mitsuru . The principal inverse of the gamma function . April 2012 . Proceedings of the American Mathematical Society. 140 . 4 . 1347 . 10.1090/S0002-9939-2011-11023-2 . 41505586 . 85549521 . free .
3. Pedersen . Henrik . "Inverses of gamma functions" . Constructive Approximation . 9 September 2013 . 7 . 2 . 251–267 . 10.1007/s00365-014-9239-1 . 1309.2167 . 253898042 .
4. Robert M.. Corless . Folitse Komla. Amenyou . Jeffrey . David . 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) . Properties and Computation of the Functional Inverse of Gamma . International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC) . 2017 . 65 . 10.1109/SYNASC.2017.00020. 978-1-5386-2626-9 . 53287687 .
5. MS . Amenyou . Folitse Komla . Jeffrey . David . "Properties and Computation of the inverse of the Gamma Function" . 2018 . 28 .
6. Couto . Ana Carolina Camargos . Jeffrey . David . Corless . Robert . November 2020 . The Inverse Gamma Function and its Numerical Evaluation . Section 8 . Maple Conference Proceedings.