Trigamma function

In mathematics, the trigamma function, denoted ψ₁(z) or ψ⁽¹⁾(z), is the second of the polygamma functions, and is defined by

${\displaystyle \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)}$.

It follows from this definition that

${\displaystyle \psi _{1}(z)={\frac {d}{dz}}\psi (z)}$

where ψ(z) is the digamma function. It may also be defined as the sum of the series

${\displaystyle \psi _{1}(z)=\sum _{n=0}^{\infty }{\frac {1}{(z+n)^{2}}},}$

making it a special case of the Hurwitz zeta function

${\displaystyle \psi _{1}(z)=\zeta (2,z).}$

Note that the last two formulas are valid when 1 − z is not a natural number.