Trigamma function


In mathematics, the trigamma function, denoted, is the second of the polygamma functions, and is defined by
It follows from this definition that
where is the digamma function. It may also be defined as the sum of the series
making it a special case of the Hurwitz zeta function
Note that the last two formulas are valid when 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:
using the formula for the sum of a geometric series. Integration over yields:
An asymptotic expansion as a Laurent series is
if we have chosen, i.e. the Bernoulli numbers of the second kind.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation
and the reflection formula
which immediately gives the value for z :.

Special values

At positive half integer values we have that
Moreover, the trigamma function has the following special values:
where represents Catalan's constant.
There are no roots on the real axis of, but there exist infinitely many pairs of roots for. Each such pair of roots approaches quickly and their imaginary part increases slowly logarithmic with. For example, and are the first two roots with.

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,

Computation and approximation

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

Appearance

The trigamma function appears in this surprising sum formula: