mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the inverse gamma function

\Gamma^(x)

is the

inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...

of the

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

. In other words,

y = \Gamma^(x)

whenever

\Gamma(y)=x

. For example,

\Gamma^(24)=5

. Usually, the inverse gamma function refers to the principal branch with domain on the real interval

\left[\beta, +\infty\right)

and image on the real interval

\left[\alpha, +\infty\right)

, where

\beta = 0.8856031\ldots

is the minimum value of the gamma function on the positive real axis and

\alpha = \Gamma^(\beta) = 1.4616321\ldots

is the location of that minimum.

Definition

The inverse gamma function may be defined by the following integral representation

\Gamma^(x)=a+bx+\int_^\left(\frac-\frac\right)d\mu(t)\,,

where

\mu (t)

is a Borel measure such that

\int_^\left(\frac\right)d\mu(t)<\infty \,,

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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

\Gamma(x)

near

\alpha

. The series can then be truncated and inverted, which yields successively better approximations to

\Gamma^(x)

. For instance, we have the quadratic approximation:

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

where

\psi^ \left(x \right)

is the

trigamma function In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by : \psi_1(z) = \frac \ln\Gamma(z). It follows from this definition that : \psi_1(z) = \frac \psi(z) where is the digamma functi ...

. The inverse gamma function also has the following asymptotic formula

\Gamma^(x)\sim\frac+\frac\,,

where

W_0(x)

is the

Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...

. 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

\frac

near the poles at the negative integers, and then invert the series. Setting

z=\frac

then yields, for the ''n'' th branch

\Gamma_^(z)

of the inverse gamma function (

n\ge 0

)

\Gamma_^(z)=-n+\frac+\frac+\frac+O\left(\frac\right)\,,

where

\psi^(x)

is the

polygamma function In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) ...

References

Gamma and related functions {{mathematics-stub