Trigamma function

File:Psi1.png

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) or ψ⁽¹⁾(z), is the second of the polygamma functions, and is defined by

${\psi }_1(z)=\frac{d^2}{d{z}^2}\ln \mathrm{\Gamma }(z)$.

It follows from this definition that

${\psi }_1(z)=\frac{d}{dz}\psi (z)$

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

${\psi }_1(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(z+n{)}^2},$

making it a special case of the Hurwitz zeta function

${\psi }_1(z)=\zeta (2,z).$

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:

${\psi }_1(z)=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{x}{\int }}\frac{x^{z-1}}{y(1-x)}dydx$

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

${\psi }_1(z)=-\underset{0}{\overset{1}{\int }}\frac{x^{z-1}\ln x}{1-x}dx$

An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:

$\begin{array}{rl}{\psi }_1(z)& \sim \frac{\mathrm{d}}{\mathrm{d}z}(\ln z-\sum _{n=1}^{\mathrm{\infty }}\frac{B_n}{n{z}^n})\\ & =\frac{1}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{B_n}{z^{n+1}}=\sum _{n=0}^{\mathrm{\infty }}\frac{B_n}{z^{n+1}}\\ & =\frac{1}{z}+\frac{1}{2z^2}+\frac{1}{6z^3}-\frac{1}{30z^5}+\frac{1}{42z^7}-\frac{1}{30z^9}+\frac{5}{66z^{11}}-\frac{691}{2730z^{13}}+\frac{7}{6z^{15}}\cdots \end{array}$

where Template:Mvar is the Template:Mvarth Bernoulli number and we choose B₁ = Template:Sfrac.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

${\psi }_1(z+1)={\psi }_1(z)-\frac{1}{z^2}$

and the reflection formula

${\psi }_1(1-z)+{\psi }_1(z)=\frac{{\pi }^2}{{\sin }^2\pi z}$

which immediately gives the value for z = Template:Sfrac: ${\psi }_1(\frac{1}{2})=\frac{{\pi }^2}{2}$.

Special values

At positive integer values we have that

${\psi }_1(n)=\frac{{\pi }^2}{6}-\sum _{k=1}^{n-1}\frac{1}{k^2},{\psi }_1(1)=\frac{{\pi }^2}{6},{\psi }_1(2)=\frac{{\pi }^2}{6}-1,{\psi }_1(3)=\frac{{\pi }^2}{6}-\frac{5}{4}.$

At positive half integer values we have that

${\psi }_1(n+\frac{1}{2})=\frac{{\pi }^2}{2}-4\sum _{k=1}^n\frac{1}{(2k-1{)}^2},{\psi }_1\left(\frac{1}{2}\right)=\frac{{\pi }^2}{2},{\psi }_1\left(\frac{3}{2}\right)=\frac{{\pi }^2}{2}-4.$

The trigamma function has other special values such as:

${\psi }_1\left(\frac{1}{4}\right)={\pi }^2+8G$

where Template:Mvar 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 + Template:Sfrac quickly and their imaginary part increases slowly logarithmic with Template:Mvar. 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,

${\psi }_1\left(\frac{p}{q}\right)=\frac{{\pi }^2}{2{\sin }^2(\pi p/q)}+2q\sum _{m=1}^{(q-1)/2}\sin \left(\frac{2\pi mp}{q}\right){\text{<mrow data-mjx-texclass="ORD">Cl</mrow>}}_2\left(\frac{2\pi m}{q}\right).$

Appearance

The trigamma function appears in this sum formula:

$\sum _{n=1}^{\mathrm{\infty }}\frac{n^2-\frac{1}{2}}{{(n^2+\frac{1}{2})}^2}\left({\psi }_1\right(n-\frac{i}{\sqrt{2}})+{\psi }_1(n+\frac{i}{\sqrt{2}}\left)\right)=-1+\frac{\sqrt{2}}{4}\pi \coth \frac{\pi }{\sqrt{2}}-\frac{3{\pi }^2}{4{\sinh }^2\frac{\pi }{\sqrt{2}}}+\frac{{\pi }^4}{12{\sinh }^4\frac{\pi }{\sqrt{2}}}(5+\cosh \pi \sqrt{2}).$

See also

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

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. Template:ISBN. See section §6.4
• Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource