In mathematics, the inverse gamma function Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gamma function. In other words, y = Γ − 1 ( x ) {\displaystyle y=\Gamma ^{-1}(x)} whenever Γ ( y ) = x {\textstyle \Gamma (y)=x} . For example, Γ − 1 ( 24 ) = 5 {\displaystyle \Gamma ^{-1}(24)=5} . Usually, the inverse gamma function refers to the principal branch with domain on the real interval [ β , + ∞ ) {\displaystyle \left[\beta ,+\infty \right)} and image on the real interval [ α , + ∞ ) {\displaystyle \left[\alpha ,+\infty \right)} , where β = 0.8856031 … {\displaystyle \beta =0.8856031\ldots } is the minimum value of the gamma function on the positive real axis and α = Γ − 1 ( β ) = 1.4616321 … {\displaystyle \alpha =\Gamma ^{-1}(\beta )=1.4616321\ldots } is the location of that minimum.
Definition The inverse gamma function may be defined by the following integral representation
Γ − 1 ( x ) = a + b x + ∫ − ∞ Γ ( α ) ( 1 x − t − t t 2 − 1 ) d μ ( t ) , {\displaystyle \Gamma ^{-1}(x)=a+bx+\int _{-\infty }^{\Gamma (\alpha )}\left({\frac {1}{x-t}}-{\frac {t}{t^{2}-1}}\right)d\mu (t)\,,}
where μ ( t ) {\displaystyle \mu (t)} is a Borel measure such that ∫ − ∞ Γ ( α ) ( 1 t 2 + 1 ) d μ ( t ) < ∞ , {\displaystyle \int _{-\infty }^{\Gamma \left(\alpha \right)}\left({\frac {1}{t^{2}+1}}\right)d\mu (t)<\infty \,,} and a {\displaystyle a} and b {\displaystyle b} are real numbers with b ≧ 0 {\displaystyle b\geqq 0} .
Series Expansions Let Γ n − 1 ( z ) {\displaystyle \Gamma _{n}^{-1}(z)} be the n {\displaystyle n} -th branch of the gamma function, with Γ 0 − 1 ( z ) {\displaystyle \Gamma _{0}^{-1}(z)} denoting the principal branch. To obtain a series expansion of the inverse gamma function one can first compute the series expansion of the reciprocal gamma function 1 Γ ( x ) {\displaystyle {\frac {1}{\Gamma (x)}}} near the zeros at the negative integers, and then invert the series. Setting z = 1 x {\displaystyle z={\frac {1}{x}}} in the expansion then yields, for n ≥ 1 {\displaystyle n\geq 1} : Γ n − 1 ( z ) = − n + 1 − ( − 1 ) n ( n − 1 ) ! z + ψ ( 0 ) ( n ) ( n − 1 ) ! 2 z 2 − ( − 1 ) n ( π 2 + 9 ψ ( 0 ) ( n ) 2 − 3 ψ ( 1 ) ( n ) ) 6 ( n − 1 ) ! 3 z 3 + O ( 1 z 4 ) , {\displaystyle \Gamma _{n}^{-1}(z)=-n+1-{\frac {\left(-1\right)^{n}}{\left(n-1\right)!z}}+{\frac {\psi ^{(0)}\left(n\right)}{\left(n-1\right)!^{2}z^{2}}}-{\frac {\left(-1\right)^{n}\left(\pi ^{2}+9\psi ^{(0)}\left(n\right)^{2}-3\psi ^{(1)}\left(n\right)\right)}{6\left(n-1\right)!^{3}z^{3}}}+O\left({\frac {1}{z^{4}}}\right)\,,} where ψ ( n ) ( x ) {\displaystyle \psi ^{(n)}(x)} is the polygamma function. This can be rigorously justified by the Lagrange inversion theorem, which says that if z = f ( w ) {\displaystyle z=f(w)} where f is analytic at a point a and f ′ ( a ) ≠ 0 {\displaystyle f'(a)\neq 0} , then the inverse w = g ( z ) {\displaystyle w=g(z)} is given by a power series
g ( z ) = a + ∑ n = 1 ∞ g n ( z − f ( a ) ) n n ! , g n = lim w → a d n − 1 d w n − 1 [ ( w − a f ( w ) − f ( a ) ) n ] . {\displaystyle g(z)=a+\sum _{n=1}^{\infty }g_{n}{\frac {(z-f(a))^{n}}{n!}},\quad g_{n}=\lim _{w\to a}{\frac {d^{n-1}}{dw^{n-1}}}\left[\left({\frac {w-a}{f(w)-f(a)}}\right)^{n}\right].}
For example, for the first non principal branch, set f ( x ) = 1 Γ ( x ) {\displaystyle f(x)={\frac {1}{\Gamma (x)}}} and a = 0 {\displaystyle a=0} such that f ( a ) = 0 {\displaystyle f(a)=0} . The formula gives
g ( z ) = ∑ n = 1 ∞ z n n ! [ lim w → 0 d n − 1 d w n − 1 Γ ( w + 1 ) n ] . {\displaystyle g(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}\left[\lim _{w\to 0}{\frac {d^{n-1}}{dw^{n-1}}}\Gamma (w+1)^{n}\right].}
Now, assuming 1 Γ ( x ) {\displaystyle {\frac {1}{\Gamma (x)}}} is invertible when restricted to an appropriate region, setting z = 1 x {\displaystyle z={\frac {1}{x}}} gives
Γ 1 − 1 ( x ) = ∑ n = 1 ∞ x − n n ! [ d n − 1 d w n − 1 Γ ( w + 1 ) n ] w = 0 . {\displaystyle \Gamma _{1}^{-1}(x)=\sum _{n=1}^{\infty }{\frac {x^{-n}}{n!}}\left[{\frac {d^{n-1}}{dw^{n-1}}}\Gamma (w+1)^{n}\right]_{w=0}.}
Approximation To compute the branches of the inverse gamma function one can first compute the Taylor series of Γ ( x ) {\displaystyle \Gamma (x)} near α {\displaystyle \alpha } . The series can then be truncated and inverted, which yields successively better approximations to Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} . For instance, we have the quadratic approximation:
Γ − 1 ( x ) ≈ α + 2 ( x − Γ ( α ) ) ψ ( 1 ) ( α ) Γ ( α ) . {\displaystyle \Gamma ^{-1}\left(x\right)\approx \alpha +{\sqrt {\frac {2\left(x-\Gamma \left(\alpha \right)\right)}{\psi ^{\left(1\right)}\left(\alpha \right)\Gamma \left(\alpha \right)}}}.} where ψ ( 1 ) ( x ) {\displaystyle \psi ^{\left(1\right)}\left(x\right)} is the trigamma function. The inverse gamma function also has the following asymptotic formula
Γ − 1 ( x ) ∼ 1 2 + ln ( x 2 π ) W 0 ( e − 1 ln ( x 2 π ) ) , {\displaystyle \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)}}\,,}
where W 0 ( x ) {\displaystyle 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.
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