It follows from this definition that
making it a special case of the Hurwitz zeta function
Note that the last two formulas are valid when 1 − z is not a natural number.
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 over y yields:
An asymptotic expansion as a Laurent series is
if we have chosen B1 = 1/, i.e. the Bernoulli numbers of the second kind.
Recurrence and reflection formulaeEdit
The trigamma function satisfies the recurrence relation
and the reflection formula
which immediately gives the value for z = 1/: .
At positive half integer values we have that
Moreover, 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/ 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.
Relation to the Clausen functionEdit
The digamma function at rational arguments can be expressed in terms of trigonometric functions and logarithm by the digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by Clausen's function. Namely,
Computation and approximationEdit
An easy method to approximate the trigamma function is to take the derivative of the series expansion of the digamma function.
The trigamma function appears in this surprising sum formula:
- Lewin, L. (editor) (1991). Structural properties of polylogarithms. American Mathematical Society. ISBN 978-0821816349.CS1 maint: extra text: authors list (link)
- Mező, István (2013). "Some infinite sums arising from the Weierstrass Product Theorem". Applied Mathematics and Computation. 219 (18): 9838–9846. doi:10.1016/j.amc.2013.03.122.
- 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 Resource