Trigamma function

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Trigamma function $\psi _{1}(z)$ in the complex plane. The color of a point $z$ encodes the value of $\psi _{1}(z)$ . Strong colors denote values close to zero and hue encodes the value's argument.

In mathematics, the trigamma function, denoted $\psi _{1}(z)$ , is the second of the polygamma functions, and is defined by

$\psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z)$ .

It follows from this definition that

$\psi _{1}(z)={\frac {d}{dz}}\psi (z)$

where $\psi (z)$ 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

$\psi _{1}(z)=\zeta (2,z).{\frac {}{}}$

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:

$\psi _{1}(z)=\int _{0}^{1}\int _{0}^{y}{\frac {x^{{z-1}}y}{1-x}}\,dx\,dy$

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

$\psi _{1}(z)=-\int _{0}^{1}{\frac {x^{{z-1}}\ln {x}}{1-x}}\,dx$

An asymptotic expansion as a Laurent series is

$\psi _{1}(z)={\frac {1}{z}}+{\frac {1}{2z^{2}}}+\sum _{{k=1}}^{{\infty }}{\frac {B_{{2k}}}{z^{{2k+1}}}}=\sum _{{k=0}}^{{\infty }}{\frac {B_{k}}{z^{{k+1}}}}$

if we have chosen $B_{1}=1/2$ , i.e. the Bernoulli numbers of the second kind.

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

$\psi _{1}(1-z)+\psi _{1}(z)={\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}\,$

which immediately gives the value for z=1/2.

Special values

The trigamma function has the following special values:

$\psi _{1}\left({\frac {1}{4}}\right)=\pi ^{2}+8K$

$\psi _{1}\left({\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}$

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

$\psi _{1}\left({\frac {3}{2}}\right)={\frac {\pi ^{2}}{2}}-4$

$\psi _{1}(2)={\frac {\pi ^{2}}{6}}-1$

where K represents Catalan's constant.

There are no roots on the real axis of $\psi _{1}$ , but there exist infinitely many pairs of roots $z_{n},\overline {z_{n}}$ for $\Re (z)<0$ . Each such pair of root approach $\Re (z_{n})=-n+1/2$ quickly and their imaginary part increases slowly logarithmic with n. E.g. $z_{1}=-0.4121345\ldots +i0.5978119\ldots$ and $z_{2}=-1.4455692\ldots +i0.6992608\ldots$ are the first two roots with $\Im (z)>0$ .

Appearance

The trigamma function appears in the next surprising sum formula:^[1]

$\sum _{{n=1}}^{\infty }{\frac {n^{2}-{\frac 12}}{\left(n^{2}+{\frac 12}\right)^{2}}}\left[\psi _{1}\left(n-{\frac {i}{{\sqrt {2}}}}\right)+\psi _{1}\left(n+{\frac {i}{{\sqrt {2}}}}\right)\right]=-1+{\frac {{\sqrt {2}}}{4}}\pi \coth \left({\frac {\pi }{{\sqrt {2}}}}\right)-{\frac {3\pi ^{2}}{4\sinh ^{2}\left({\frac {\pi }{{\sqrt {2}}}}\right)}}+{\frac {\pi ^{4}}{12\sinh ^{4}\left({\frac {\pi }{{\sqrt {2}}}}\right)}}\left(5+\cosh \left(\pi {\sqrt {2}}\right)\right).$

Notes

↑ 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

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

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