Trigamma function

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For Barnes's gamma function of 3 variables, see triple gamma function.
Color representation of the Trigamma function, , in a rectangular region of the complex plane. It is generated using Domain coloring method.

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 formulæ 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 by parts 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[edit]

The trigamma function satisfies the recurrence relation

and the reflection formula

which immediately gives the value for z=1/2.

Special values[edit]

The trigamma function has the following special values:

where K 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 root approach quickly and their imaginary part increases slowly logarithmic with n. E.g. and are the first two roots with .


The trigamma function appears in the next surprising sum formula:[1]

See also[edit]


  1. ^ Mező, István (2013). "Some infinite sums arising from the Weierstrass Product Theorem". Applied Mathematics and Computation 219: 9838–9846. doi:10.1016/j.amc.2013.03.122.