Trigamma function

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

: $psi\_1(z)\; =\; frac\{d^2\}\{dz^2\}\; lnGamma(z)$.

It follows from this definition that

: $psi\_1(z)\; =\; frac\{d\}\{dz\}\; psi(z)$

where ψ(z) is the digamma function. It may also be defined as the sum of the series

: $psi\_1(z)\; =\; sum\_\{n\; =\; 0\}^\{infty\}frac\{1\}\{(z\; +\; n)^2\},$

making it a special case of the Hurwitz zeta function

: $psi\_1(z)\; =\; zeta(2,z).$

Note that the last two formulæ are valid when 1-"z" 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:

: $psi\_1(z)\; =\; int\_0^1frac\{dy\}\{y\}int\_0^yfrac\{x^\{z-1\},dx\}\{1\; -\; x\}$

using the formula for the sum of a geometric series. Integration by parts yields:

: $psi\_1(z)\; =\; -int\_0^1frac\{x^\{z-1\}ln\{x\{1-x\},dx$

An asymptotic expansion in terms of the Bernoulli numbers is

$psi\_1(z)\; sim\; frac\{1\}\{z\}\; +\; frac\{1\}\{2z^2\}\; +\; sum\_\{k=1\}^\{infty\}frac\{B\_\{2k\{z^\{2k+1$.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation:

: $psi\_1(z\; +\; 1)\; =\; psi\_1(z)\; -\; frac\{1\}\{z^2\}$

and the reflection formula:

: $psi\_1(1\; -\; z)\; +\; psi\_1(z)\; =\; pi^2csc^2(pi\; z).\; ,$

pecial values

The trigamma function has the following special values:

$psi\_1left(frac\{1\}\{4\}\; ight)\; =\; pi^2\; +\; 8K$

$psi\_1left(frac\{1\}\{2\}\; ight)\; =\; frac\{pi^2\}\{2\}$

$psi\_1(1)\; =\; frac\{pi^2\}\{6\}$

where K represents Catalan's constant.

ee also

* Gamma function
* Digamma function
* Polygamma function
* Catalan's constant

References

* Milton Abramowitz and Irene A. Stegun, "Handbook of Mathematical Functions", (1964) Dover Publications, New York. ISBN 0-486-61272-4. See section [http://www.math.sfu.ca/~cbm/aands/page_260.htm §6.4]
* Eric W. Weisstein. [http://mathworld.wolfram.com/TrigammaFunction.html Trigamma Function -- from MathWorld--A Wolfram Web Resource]

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