Trigamma function
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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
- ψ1(z) = ζ(2,z).
Note that the last two formulæ are valid when 1-z is not a natural number.
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[edit] 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 by parts yields:
An asymptotic expansion in terms of the Bernoulli numbers is
.
[edit] Recurrence and reflection formulae
The trigamma function satisfies the recurrence relation:
and the reflection formula:
[edit] Special values
The trigamma function has the following special values:
where K represents Catalan's constant.
[edit] See also
[edit] References
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. See section §6.4
- Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resourceæ
