logGamma

logGamma(x): number

Defined in: packages/math-analysis/src/log-gamma.ts:32

The natural logarithm of the gamma function, ln Γ(x).

Use it whenever Γ itself would not fit. Γ(a) overflows binary64 past a ≈ 171.6, while ln Γ(a) stays finite for every argument a caller can represent — which is what makes the ratios of gamma functions behind beta and the probability distributions computable at all.

Defined for x > 0 only. That is not a gap: Γ alternates sign between consecutive negative integers and has a pole at each of them, so no real logarithm covers the negative reals. Use gamma there, which carries the sign.

O(1) time, O(1) memory — a fixed nine-term series.

Parameters

x

number

A positive real.

Returns

number

ln Γ(x), or NaN when x ≤ 0 or x is NaN.

Example

  logGamma(172); // 711.71472580229 — finite where gamma(172) is Infinity
  logGamma(1e6); // 12815504.56914761
  logGamma(-1); // NaN — no real logarithm on the negative reals
  ```;

## See

 - [gamma](gamma.md) for the value itself, and for negative arguments.
 - [The gamma and beta functions](https://anthochamp.pages.grenadine.io/node-essentials/topics/math/functions/gamma/)