Trigamma function

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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