Lanczos approximation: Difference between revisions

Revision as of 18:29, 25 June 2019

Revision as of 14:50, 5 November 2019

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The Lanczos approximation consists of the formula

:<math>\Gamma(z+1) = \sqrt{2\pi} {\left( z + g + \tfrac12 \right)}^{z + \frac121/2 } e^{-\left(z+g+\frac12\right1/2)} A_g(z)</math>

for the gamma function, with

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==Coefficients==

The coefficients are given by

:<math>p_k(g) = \frac{\sqrt{2\,}}{\pi} \sum_{\ell=0}^k C_{2k+1,\,2\ell+1} \left(\ell - \tfrac{1}{2} \right)! {\left(\ell + g + \tfrac{1}{2} \right)}^{- \left( \ell + \tfrac{1}{/2} \right) } e^{\ell + g + \frac121/2 }</math>

where <math>C_{n,m}</math> represents the (''n'', ''m'')th element of the [[matrix (mathematics)|matrix]] of coefficients for the [[Chebyshev polynomial]]s, which can be calculated [[recursion|recursively]] from these identities: