digammafunktioiden

The digamma function, denoted by $\psi(x)$, is the logarithmic derivative of the gamma function. Mathematically, it is defined as $\psi(x) = \frac{d}{dx} \ln(\Gamma(x))$, where $\Gamma(x)$ is the gamma function. This means that if $\Gamma(x) = \int_0^\infty t^{x-1}e^{-t} dt$, then $\psi(x) = \frac{\Gamma'(x)}{\Gamma(x)}$.

The digamma function has several important properties. For positive integers $n$, the digamma function can be expressed as $\psi(n) = -\gamma + \sum_{k=1}^{n-1} \frac{1}{k}$, where $\gamma$ is the Euler-Mascheroni constant. It also satisfies the recurrence relation $\psi(x+1) = \psi(x) + \frac{1}{x}$. Another key identity is the reflection formula: $\psi(1-x) - \psi(x) = \pi \cot(\pi x)$.

The digamma function appears in various fields of mathematics and physics, including probability theory, statistics, and number theory. It is particularly relevant in the study of distributions such as the Beta distribution and the Gamma distribution. Its series representation is given by $\psi(x) = -\gamma - \frac{1}{x} - \sum_{n=1}^\infty \left(\frac{1}{x+n} - \frac{1}{n}\right)$. The digamma function is also the first of the polygamma functions, with the $m$-th polygamma function being the $m$-th derivative of the digamma function.