Trigamma function

In mathematics, the trigamma function, denoted $ψ 1 (z)$ or $ψ (1) (z)$ , is the second of the polygamma functions, and is defined by

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

.

Color representation of the trigamma function,

ψ 1 (z)

, in a rectangular region of the complex plane. It is generated using the domain coloring method.

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 formulas 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}^{x}{\frac {x^{z-1}}{y(1-x)}}\,dy\,dx

using the formula for the sum of a geometric series. Integration over $y$ 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}}}

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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: $\psi _{1}({\tfrac {1}{2}})={\tfrac {\pi ^{2}}{2}}$ .

Special values

At positive half integer values we have that

\psi _{1}\left(n+{\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}-4\sum _{k=1}^{n}{\frac {1}{(2k-1)^{2}}}.

Moreover, the trigamma function has the following special values:

{\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\quad &\psi _{1}\left({\tfrac {1}{2}}\right)&={\frac {\pi ^{2}}{2}}&\psi _{1}(1)&={\frac {\pi ^{2}}{6}}\\[6px]\psi _{1}\left({\tfrac {3}{2}}\right)&={\frac {\pi ^{2}}{2}}-4&\psi _{1}(2)&={\frac {\pi ^{2}}{6}}-1\quad \end{aligned}}

where $G$ represents Catalan's constant.

There are no roots on the real axis of $ψ 1$ , but there exist infinitely many pairs of roots $z n, z n$ for $Re z < 0$ . Each such pair of roots approaches $Re z n = - n + 1 / 2$ quickly and their imaginary part increases slowly logarithmic with $n$ . For example, $z 1 = -0.4121345... + 0.5978119... i$ and $z 2 = -1.4455692... + 0.6992608... i$ are the first two roots with $Im(z) > 0$ .

Relation to the Clausen function

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,[1]

\psi _{1}\left({\frac {p}{q}}\right)={\frac {\pi ^{2}}{2\sin ^{2}(\pi p/q)}}+2q\sum _{m=1}^{(q-1)/2}\sin \left({\frac {2\pi mp}{q}}\right){\textrm {Cl}}_{2}\left({\frac {2\pi m}{q}}\right).

Computation and approximation

An easy method to approximate the trigamma function is to take the derivative of the asymptotic expansion of the digamma function.

\psi _{1}(x)\approx {\frac {1}{x}}+{\frac {1}{2x^{2}}}+{\frac {1}{6x^{3}}}-{\frac {1}{30x^{5}}}+{\frac {1}{42x^{7}}}-{\frac {1}{30x^{9}}}+{\frac {5}{66x^{11}}}-{\frac {691}{2730x^{13}}}+{\frac {7}{6x^{15}}}

Appearance

The trigamma function appears in this sum formula:[2]

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

Notes

Lewin, L., ed. (1991). Structural properties of polylogarithms. American Mathematical Society. ISBN 978-0821816349.
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.

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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[1] Lewin, L., ed. (1991). Structural properties of polylogarithms. American Mathematical Society. ISBN 978-0821816349.

[mezo-2] 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.