Английская Википедия:Inverse gamma function
Шаблон:Short description Шаблон:Distinguish Шаблон:Orphan Шаблон:Multiple image
In mathematics, the inverse gamma function <math>\Gamma^{-1}(x)</math> is the inverse function of the gamma function. In other words, <math>y = \Gamma^{-1}(x)</math> whenever <math display="inline">\Gamma(y)=x</math>. For example, <math>\Gamma^{-1}(24)=5</math>.[1] Usually, the inverse gamma function refers to the principal branch with domain on the real interval <math>\left[\beta, +\infty\right)</math> and image on the real interval <math>\left[\alpha, +\infty\right)</math>, where <math>\beta = 0.8856031\ldots</math>[2] is the minimum value of the gamma function on the positive real axis and <math>\alpha = \Gamma^{-1}(\beta) = 1.4616321\ldots</math>[3] is the location of that minimum.[4]
Definition
The inverse gamma function may be defined by the following integral representation[5] <math display="block">\Gamma^{-1}(x)=a+bx+\int_{-\infty}^{\Gamma(\alpha)}\left(\frac{1}{x-t}-\frac{t}{t^{2}-1}\right)d\mu(t)\,,</math> where <math>\mu (t)</math> is a Borel measure such that <math display="block">\int_{-\infty}^{\Gamma\left(\alpha\right)}\left(\frac{1}{t^{2}+1}\right)d\mu(t)<\infty \,,</math> and <math>a</math> and <math>b</math> are real numbers with <math>b \geqq 0</math>.
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:[6]
<math display="block"> \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[7] <math display="block"> \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> where <math>W_0(x)</math> 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 <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>)[8] <math display="block"> \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> where <math>\psi^{(n)}(x)</math> is the polygamma function.
References
Шаблон:Reflist Шаблон:Mathematics-stub
