In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

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

for complex numbers with large modulus (${\displaystyle |z|\rightarrow \infty }$) in the sector ${\displaystyle |\arg z|<\pi -\varepsilon }$ for any ${\displaystyle \varepsilon >0}$.

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

The gamma function obeys the equation

Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives:

Differentiating both sides with respect to z gives:

Since the harmonic numbers are defined for positive integers n as