Degamma

Degamma, also known as the inverse gamma function or the inverse logarithmic gamma function, is a mathematical function that reverses the operation of the gamma function. The gamma function, denoted by $\Gamma(z)$, is a generalization of the factorial function to complex and real numbers. It is defined for complex numbers $z$ with a positive real part by the integral $\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt$. The digamma function, denoted by $\psi(z)$, is the logarithmic derivative of the gamma function, meaning $\psi(z) = \frac{d}{dz} \ln \Gamma(z)$.

Degamma is the inverse of the digamma function. If $\psi(x) = y$, then degamma, often denoted by $\psi^{-1}(y)$, would return $x$. Therefore, $\psi(\psi^{-1}(y)) = y$ and $\psi^{-1}(\psi(x)) = x$. Unlike the gamma function itself, which has an infinite number of roots, the digamma function is strictly increasing for real arguments greater than zero, which ensures that its inverse, degamma, is well-defined for these values.

The degamma function does not have a simple closed-form expression in terms of elementary functions. Its values are typically computed using numerical methods or series expansions. It appears in various areas of mathematics and statistics, including Bayesian statistics, probability theory, and the analysis of certain combinatorial problems. For instance, it arises when working with distributions related to the gamma distribution.