Trigamma function

Trigamma function

ψ 1 (z)

in the complex plane. The color of a point

z

encodes the value of

ψ 1 (z)

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

In mathematics, the trigamma function, denoted ψ₁(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 ψ(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

ψ 1 (z) = ζ(2, z).

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

[edit] Calculation

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

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

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 in terms of the Bernoulli numbers is

$\psi_1(z) \sim \frac{1}{z} + \frac{1}{2z^2} + \sum_{k=1}^{\infty}\frac{B_{2k}}{z^{2k+1}}$ .

The trigamma function satisfies the recurrence relation:

$\psi_1(z + 1) = \psi_1(z) - \frac{1}{z^2}$

$\psi_1(1 - z) + \psi_1(z) = \pi^2\csc^2(\pi z). \,$

The trigamma function has the following special values:

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

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

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

where K represents Catalan's constant.

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. See section §6.4
Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resourceæ