Gamma Function

The Gamma function $\Gamma(n)$ generalizes the factorial to real (and complex) numbers:

$\Gamma(n) = \int_0^\infty t^{n-1} e^{-t}\, dt$

For positive integers: $\Gamma(n) = (n-1)!$

Key Properties

• Recurrence: $\Gamma(n+1) = n\, \Gamma(n)$
• Half-integer: $\Gamma(1/2) = \sqrt{\pi}$, $\Gamma(3/2) = \frac{1}{2}\sqrt{\pi}$
• Reflection formula: $\Gamma(x)\, \Gamma(1-x) = \frac{\pi}{\sin(\pi x)}$

Stirling's Approximation

For large $n$, the log-gamma function is approximated by:

$\ln \Gamma(n) \approx \left(n - \tfrac{1}{2}\right) \ln n - n + \tfrac{1}{2} \ln(2\pi)$

This is the basis for log_gamma(n).

Lanczos Approximation

The Lanczos approximation computes $\Gamma(z)$ accurately for positive reals using a set of precomputed coefficients. For $n \geq 0.5$, let $z = n - 1$:

$\Gamma(z+1) = \sqrt{2\pi}\, t^{z+0.5}\, e^{-t}\, A_g(z)$

where $t = z + g + 0.5$ and:

$A_g(z) = c_0 + \sum_{k=1}^{g+1} \frac{c_k}{z + k}$

For $n < 0.5$, use the reflection formula:

$\Gamma(n) = \frac{\pi}{\sin(\pi n)\, \Gamma(1-n)}$

Beta Function

The Beta function is defined as:

$B(a, b) = \frac{\Gamma(a)\, \Gamma(b)}{\Gamma(a+b)} = \int_0^1 t^{a-1}(1-t)^{b-1}\, dt$

It arises in probability (Beta distribution), combinatorics, and integration.

Your Task

Implement gamma using the Lanczos approximation with $g=7$ and the 9-coefficient vector below. Implement log_gamma using Stirling's approximation. Implement beta_function using the Gamma function.

Lanczos coefficients (g=7):

[0.99999999999980993, 676.5203681218851, -1259.1392167224028,
 771.32342877765313, -176.61502916214059, 12.507343278686905,
 -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]