Digamma function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln ⁡ Γ ( z ) = Γ ′ ( z ) Γ ( z ) . {\displaystyle \psi (z)={\frac {d}{dz}}\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 ( 0 , ∞ ) {\displaystyle (0,\infty )} , and it asymptotically behaves as ψ ( z ) ∼ ln ⁡ z − 1 2 z , {\displaystyle \psi (z)\sim \ln {z}-{\frac {1}{2z}},} for complex numbers with large modulus ( | z | → ∞ {\displaystyle |z|\rightarrow \infty } ) in the sector | arg ⁡ z | < π − ε {\displaystyle \left|\arg z\right|<\pi -\varepsilon } for any ε > 0 {\displaystyle \varepsilon >0} . The digamma function is often denoted as ψ 0 ( x ) , ψ ( 0 ) ( x ) {\displaystyle \psi _{0}(x),\psi ^{(0)}(x)} or Ϝ (the uppercase form of the archaic Greek letter digamma meaning double-gamma).