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Wikipedia / Trigamma function

In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by

  1. .

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 can be obtained via the derivative of the asymptotic expansion of the digamma function:

where is the th Bernoulli number and we choose .

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 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 .

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,

Appearance

The trigamma function appears in this sum formula:

See also

Notes

References