Trigamma function

For Barnes's gamma function of 3 variables, see triple gamma function.

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

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

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

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:

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

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 can be obtained via the derivative of the asymptotic expansion of the digamma function:

${\displaystyle {\begin{aligned}\psi _{1}(z)&\sim {\operatorname {d} \over \operatorname {d} \!z}\left(\ln z-\sum _{n=1}^{\infty }{\frac {B_{n}}{nz^{n}}}\right)\\&={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {B_{n}}{z^{n+1}}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{z^{n+1}}}\\&={\frac {1}{z}}+{\frac {1}{2z^{2}}}+{\frac {1}{6z^{3}}}-{\frac {1}{30z^{5}}}+{\frac {1}{42z^{7}}}-{\frac {1}{30z^{9}}}+{\frac {5}{66z^{11}}}-{\frac {691}{2730z^{13}}}+{\frac {7}{6z^{15}}}\cdots \end{aligned}}}$

where B_n is the nth Bernoulli number and we choose B₁ = ⁠1/2⁠.

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

At positive integer values we have that

${\displaystyle \psi _{1}(n)={\frac {\pi ^{2}}{6}}-\sum _{k=1}^{n-1}{\frac {1}{k^{2}}},\qquad \psi _{1}(1)={\frac {\pi ^{2}}{6}},\qquad \psi _{1}(2)={\frac {\pi ^{2}}{6}}-1,\qquad \psi _{1}(3)={\frac {\pi ^{2}}{6}}-{\frac {5}{4}}.}$

At positive half integer values we have that

${\displaystyle \psi _{1}\left(n+{\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}-4\sum _{k=1}^{n}{\frac {1}{(2k-1)^{2}}},\qquad \psi _{1}\left({\tfrac {1}{2}}\right)={\frac {\pi ^{2}}{2}},\qquad \psi _{1}\left({\tfrac {3}{2}}\right)={\frac {\pi ^{2}}{2}}-4.}$

The trigamma function has other special values such as:

${\displaystyle \psi _{1}\left({\tfrac {1}{4}}\right)=\pi ^{2}+8G}$

where G represents Catalan's constant.

There are no roots on the real axis of ψ₁, but there exist infinitely many pairs of roots z_n, z_n for Re z < 0. Each such pair of roots approaches Re z_n = −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]

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

Appearance

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

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

See also

• Gamma function
• Digamma function
• Polygamma function
• Catalan's constant