Lanczos approximation

In mathematics, the Lanczos approximation is an approximation of the Gamma function published in 1964 by Cornelius Lanczos.

Definition

Lanczos gives the approximation as

$\Gamma (z+1)=z!={\sqrt {2\pi }}{\left(z+g+{\begin{matrix}{\frac {1}{2}}\end{matrix}}\right)}^{z+{\frac {1}{2}}}e^{-\left(z+g+{\frac {1}{2}}\right)}A(z)$,

for an arbitrary positive constant g, with

$A(z)={\frac {1}{2}}p_{0}+p_{1}{\frac {z}{z+1}}+p_{2}{\frac {z(z-1)}{(z+1)(z+2)}}+...$

or equivalently,

$A(z)={\frac {1}{2}}p_{0}+p_{1}\left(1-{\frac {1}{z+1}}\right)+p_{2}\left(1+{\frac {2}{z+1}}-{\frac {6}{z+3}}\right)+...$.

Here,

$p_{k}=\sum _{a=0}^{k}C(2k+1,2a+1){\frac {\sqrt {2}}{\pi }}\left(a-{\begin{matrix}{\frac {1}{2}}\end{matrix}}\right)!{\left(a+g+{\begin{matrix}{\frac {1}{2}}\end{matrix}}\right)}^{-\left(a+{\frac {1}{2}}\right)}e^{a+g+{\frac {1}{2}}}$

with $C(i,j)$ denoting the (i, j)th element of the Chebyshev polynomial coefficient matrix which can be calculated recursively from the identities

$C(1,1)=1$
$C(2,2)=1$
$C(i,1)=-C(i-2,j)$	$i=3,4,...$
$C(i,j)=2C(i-1,j-1)-C(i-2,j)$	$i=3,4...;\;j=2,3,...$.

Lanczos' approximation is only accurate for the right complex half plane, but can be extended to the entire complex plane by the reflection formula,

$\Gamma (1-z)\;\Gamma (z)={\pi \over \sin \pi z}$.

Implementations

The Lanczos approximation for the Gamma function is used in the GNU Scientific Library.

References

• Lanczos, C (1964). "A Precision Approximation of the Gamma Function". SIAM Journal on Numerical Analysis series B, volume 1.
• Godfrey, P (2001). Lanczos Implementation of the Gamma Function

External links

• Weisstein, Eric W. "Lanczos Approximation". MathWorld.