Trigamma function

Trigamma function

\psi_1(z)

in the complex plane. The color of a point

z

encodes the value of

\psi_1(z)

. Strong colors denote values close to zero and hue encodes the value's argument.

In mathematics, the trigamma function, denoted $\psi_1(z)$ , is the second of the polygamma functions, and is defined by

\psi_1(z) = \frac{d^2}{dz^2} \ln\Gamma(z)

It follows from this definition that

\psi_1(z) = \frac{d}{dz} \psi(z)

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

\psi_1(z) = \sum_{n = 0}^{\infty}\frac{1}{(z + n)^2},

making it a special case of the Hurwitz zeta function

\psi_1(z) = \zeta(2,z). \frac{}{}

Note that the last two formulæ 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) = \int_0^1\int_0^y\frac{x^{z-1}y}{1 - x}\,dx\,dy

using the formula for the sum of a geometric series. Integration by parts yields:

\psi_1(z) = -\int_0^1\frac{x^{z-1}\ln{x}}{1-x}\,dx

An asymptotic expansion as a Laurent series is

\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 $B_1 = 1/2$ , i.e. the Bernoulli numbers of the second kind.

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=1/2.

Special values

The trigamma function has the following special values:

\psi_1\left(\frac{1}{4}\right) = \pi^2 + 8K

\psi_1\left(\frac{1}{2}\right) = \frac{\pi^2}{2}

\psi_1(1) = \frac{\pi^2}{6}

\psi_1\left(\frac{3}{2}\right) = \frac{\pi^2}{2} - 4

\psi_1(2) = \frac{\pi^2}{6} - 1

where K represents Catalan's constant.

There are no roots on the real axis of $\psi_1$ , but there exist infinitely many pairs of roots $z_n, \overline{z_n}$ for $\Re(z) < 0$ . Each such pair of root approach $\Re(z_n)= -n+1/2$ quickly and their imaginary part increases slowly logarithmic with n. E.g. $z_1 = -0.4121345\ldots + i 0.5978119\ldots$ and $z_2 = -1.4455692\ldots + i 0.6992608\ldots$ are the first two roots with $\Im(z) > 0$ .