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.

Calculation[edit]

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 .

Appearance[edit]

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

See also[edit]

Notes[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. 

References[edit]