# Trigamma function

Color representation of the trigamma function, ψ1(z), in a rectangular region of the complex plane. It is generated using the domain coloring method.

In mathematics, the trigamma function, denoted ψ1(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.

## Calculation

A double integral representation, as an alternative to the ones given above, may be derived from the series representation:

${\displaystyle \psi _{1}(z)=\int _{0}^{1}\!\!\int _{0}^{x}{\frac {x^{z-1}}{y(1-x)}}\,dx\,dy}$

using the formula for the sum of a geometric series. Integration over y yields:

${\displaystyle \psi _{1}(z)=-\int _{0}^{1}{\frac {x^{z-1}\ln {x}}{1-x}}\,dx}$

An asymptotic expansion as a Laurent series is

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

if we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.

### Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

${\displaystyle \psi _{1}(z+1)=\psi _{1}(z)-{\frac {1}{z^{2}}}}$

and the reflection formula

${\displaystyle \psi _{1}(1-z)+\psi _{1}(z)={\frac {\pi ^{2}}{\sin ^{2}\pi z}}\,}$

which immediately gives the value for z = 1/2: ${\displaystyle \psi _{1}({\tfrac {1}{2}})={\tfrac {\pi ^{2}}{2}}}$.

### Special values

The trigamma function has the following special values:

{\displaystyle {\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\quad &\psi _{1}\left({\tfrac {1}{2}}\right)&={\frac {\pi ^{2}}{2}}&\psi _{1}(1)&={\frac {\pi ^{2}}{6}}\\[6px]\psi _{1}\left({\tfrac {3}{2}}\right)&={\frac {\pi ^{2}}{2}}-4&\psi _{1}(2)&={\frac {\pi ^{2}}{6}}-1\quad \end{aligned}}}

where G represents Catalan's constant.

There are no roots on the real axis of ψ1, but there exist infinitely many pairs of roots zn, zn for Re z < 0. Each such pair of roots approaches Re zn = −n + 1/2 quickly and their imaginary part increases slowly logarithmic with n. For example, z1 = −0.4121345... + 0.5978119...i and z2 = −1.4455692... + 0.6992608...i are the first two roots with Im(z) > 0.

### Relation to the Clausen function

The digamma function at rational arguments can be expressed in terms of trigonometric functions and logarithm by the digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by Clausen's function. Namely,[1]

${\displaystyle \psi _{1}\left({\frac {p}{q}}\right)={\frac {\pi ^{2}}{2\sin ^{2}(\pi p/q)}}+2q\sum _{m=1}^{(q-1)/2}\sin \left({\frac {2\pi mp}{q}}\right){\textrm {Cl}}_{2}\left({\frac {2\pi m}{q}}\right).}$

### Computation and approximation

An easy method to approximate the trigamma function is to take the derivative of the series expansion of the digamma function.

${\displaystyle \psi _{1}(x)\approx {\frac {1}{x}}+{\frac {1}{2x^{2}}}+{\frac {1}{6x^{3}}}-{\frac {1}{30x^{5}}}+{\frac {1}{42x^{7}}}-{\frac {1}{30x^{9}}}+{\frac {5}{66x^{11}}}-{\frac {691}{2730x^{13}}}+{\frac {7}{6x^{15}}}}$

## Appearance

The trigamma function appears in this surprising sum formula:[2]

${\displaystyle \sum _{n=1}^{\infty }{\frac {n^{2}-{\frac {1}{2}}}{\left(n^{2}+{\frac {1}{2}}\right)^{2}}}\left(\psi _{1}{\bigg (}n-{\frac {i}{\sqrt {2}}}{\bigg )}+\psi _{1}{\bigg (}n+{\frac {i}{\sqrt {2}}}{\bigg )}\right)=-1+{\frac {\sqrt {2}}{4}}\pi \coth {\frac {\pi }{\sqrt {2}}}-{\frac {3\pi ^{2}}{4\sinh ^{2}{\frac {\pi }{\sqrt {2}}}}}+{\frac {\pi ^{4}}{12\sinh ^{4}{\frac {\pi }{\sqrt {2}}}}}\left(5+\cosh \pi {\sqrt {2}}\right).}$