loggamma

LogGamma denotes the natural logarithm of the gamma function, written as log Γ(z) or sometimes lgamma. It is a complex-valued function defined for complex numbers z with Γ(z) ≠ 0, and for real arguments z > 0 it reduces to the natural logarithm of the gamma function, log Γ(z) = ln Γ(z). For positive integers n, log Γ(n) = ln (n-1)!.

Analytic structure: Γ has simple poles at nonpositive integers, so log Γ is analytic on the complex plane minus those poles, with a standard branch of the logarithm chosen on the complement of a branch cut. The derivative of log Γ is the digamma function ψ(z) = Γ'(z)/Γ(z); higher derivatives are the polygamma functions ψ^(n)(z).

Important identities include the recurrence log Γ(z+1) = log z + log Γ(z) and the reflection formula log Γ(z) + log Γ(1−z) = log π − log sin(π z). These, together with multiplication formulas, underpin many theoretical and computational properties of log Γ.

Asymptotics and computation: Stirling-type expansions give log Γ(z) ~ (z−1/2) log z − z + (1/2) log(2π) + ∑ B_{2k}/(2k(2k−1) z^{2k−1}) for large |z|. The Lanczos approximation provides practical evaluation with good accuracy across the complex plane. In statistics and combinatorics, log Γ is used to compute log factorials and log-likelihoods to avoid overflow and improve numerical stability.

In programming, functions named loggamma or lgamma appear in many libraries; care is needed for complex arguments regarding branch choices and numerical stability.