Lanczos Approximation

A method of approximating gamma function.

\[\begin{eqnarray} \Gamma(z + 1) & = & (z + r + \frac{1}{2})^{z + \frac{1}{2}} \exp \left( - \left( z + r + \frac{1}{2} \right) \right) \sqrt{2} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^{2z}\theta \left( \frac{ a_{0}(r) }{ 2 } + \sum_{k=0}^{\infty} a_{k}(r) \cos(2k\theta) \right) \ d\theta . \end{eqnarray}\]

Lemma 1

\[z \in \mathbb{C}, \ \mathrm{Re}(z) > -1, \ \Gamma(z + 1) = \alpha^{z + 1} \int_{0}^{\infty} t^{z}e^{-\alpha t} \ dt .\]

proof

$\Box$

Reference

• Lanczos approximation - Wikipedia
• Pugh, G. R. (1999). An Analysis of the Lanczos Gamma a Thesis Submitted in Partial Fulfillment of, (November 2004).