---
tag: JT95B44G4G
kind: concept
name: Log-gamma function
msc: [33B15]
sources:
  - url: https://en.wikipedia.org/wiki/Gamma_function
---

## Summary

The log-gamma function is the logarithm of the [[gamma function|0H5SYBSED9]]: $\ln\Gamma(x)$ for real $x > 0$, and its holomorphic extension $\operatorname{Log}\Gamma$ to the slit plane $\C \setminus (-\infty, 0]$.
It is used because $\Gamma$ grows faster than any exponential, so its logarithm is easier to compute and to estimate.
Its derivative is the [[digamma function|HWRRXB0YXS]], and its asymptotic expansion is [[Stirling's series|K29QYWB6A5]].

## Explanations

### L1 human

The gamma function grows so fast that its values overflow quickly; for example $\Gamma(171)$ is already larger than $10^{306}$.
Taking the logarithm turns this huge growth into a gentle one, roughly like $x \ln x$.
Products of gamma values become sums of log-gamma values, which is often simpler.
For example $\ln\Gamma(1) = \ln\Gamma(2) = 0$, and $\ln\Gamma(\tfrac12) = \tfrac12 \ln \pi$.