In mathematics, the trigamma function, denoted or , 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.
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 can be obtained via the derivative of the asymptotic expansion of the digamma function:
where is the th Bernoulli number and we choose .
The trigamma function satisfies the recurrence relation
and the reflection formula
which immediately gives the value for z : .
At positive integer values we have that
At positive half integer values we have that
The trigamma function has other special values such as:
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 .
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,
The trigamma function appears in this sum formula: