Trigamma function

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Color representation of the trigamma function, ψ1(z), in a rectangular region of the complex plane. It is generated using the domain coloring method.

In mathematics, the trigamma function, denoted ψ1(z), is the second of the polygamma functions, and is defined by

.

It follows from this definition that

where ψ(z) 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 formulae are valid when 1 − z 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 B1 = 1/2, 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 1/z = 1/2.

Special values[edit]

The trigamma function has the following special values:

where G represents Catalan's constant.

There are no roots on the real axis of ψ1, but there exist infinitely many pairs of roots zn, zn for Re z < 0. Each such pair of roots approaches Re zn = −n + 1/2 quickly and their imaginary part increases slowly logarithmic with n. For example, z1 = −0.4121345... + 0.5978119...i and z2 = −1.4455692... + 0.6992608...i are the first two roots with Im(z) > 0.

Appearance[edit]

The trigamma function appears in this 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]