Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:^[1][2][3]

${\displaystyle \psi (z)={\frac {\mathrm {d} }{\mathrm {d} z}}\ln \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}.}$

It is the first of the polygamma functions. This function is strictly increasing and strictly concave on ${\displaystyle (0,\infty )}$,^[4] and it asymptotically behaves as^[5]

${\displaystyle \psi (z)\sim \ln {z}-{\frac {1}{2z}},}$

for complex numbers with large modulus (${\displaystyle |z|\rightarrow \infty }$) in the sector ${\displaystyle |\arg z|<\pi -\varepsilon }$ with some infinitesimally small positive constant ${\displaystyle \varepsilon }$.

The digamma function is often denoted as ${\displaystyle \psi _{0}(x),\psi ^{(0)}(x)}$ or Ϝ^[6] (the uppercase form of the archaic Greek consonant digamma meaning double-gamma).