right|thumb|300px|Color representation of the trigamma function, , in a rectangular region of the complex plane. It is generated using the [[domain coloring method.]]
In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by
: <math>\psi_1(z) = \frac{d^2}{dz^2} \ln\Gamma(z)</math>.
It follows from this definition that
: <math>\psi_1(z) = \frac{d}{dz} \psi(z)</math>
where is the digamma function. It may also be defined as the sum of the series
: <math> \psi_1(z) = \sum_{n = 0}^{\infty}\frac{1}{(z + n)^2}, </math>
making it a special case of the Hurwitz zeta function
: <math> \psi_1(z) = \zeta(2,z).</math>
Note that the last two formulas are valid when 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:
: <math> \psi_1(z) = \int_0^1\!\!\int_0^x\frac{x^{z-1{y(1 - x)}\,dy\,dx</math>
using the formula for the sum of a geometric series. Integration over yields:
: <math> \psi_1(z) = -\int_0^1\frac{x^{z-1}\ln{x{1-x}\,dx </math>
An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:
:<math>\begin{align}
\psi_1(z)
&\sim {\operatorname{d}\over\operatorname{d}\!z} \left(\ln z - \sum_{n=1}^\infty \frac{B_n}{nz^n}\right) \\
&= \frac{1}{z} + \sum_{n=1}^\infty \frac{B_n}{z^{n+1 = \sum_{n=0}^{\infty}\frac{B_n}{z^{n+1 \\
&= \frac{1}{z} + \frac{1}{2z^2} + \frac{1}{6z^3} - \frac{1}{30z^5} + \frac{1}{42z^7} - \frac{1}{30z^9} + \frac{5}{66z^{11 - \frac{691}{2730z^{13 + \frac{7}{6z^{15 \cdots
\end{align}</math>
where is the th Bernoulli number and we choose .
Recurrence and reflection formulae
The trigamma function satisfies the recurrence relation
: <math> \psi_1(z + 1) = \psi_1(z) - \frac{1}{z^2}</math>
and the reflection formula
: <math> \psi_1(1 - z) + \psi_1(z) = \frac{\pi^2}{\sin^2 \pi z} \,</math>
which immediately gives the value for z : <math> \psi_1(\tfrac{1}{2})=\tfrac{\pi^2}{2} </math>.
Special values
At positive integer values we have that
:<math>
\psi_1(n) = \frac{\pi^2}{6} - \sum_{k=1}^{n-1} \frac{1}{k^2}, \qquad \psi_1(1) = \frac{\pi^2}{6}, \qquad \psi_1(2) = \frac{\pi^2}{6} - 1, \qquad \psi_1(3) = \frac{\pi^2}{6} - \frac{5}{4}.
</math>
At positive half integer values we have that
:<math>
\psi_1\left(n+\frac12\right)=\frac{\pi^2}{2}-4\sum_{k=1}^n\frac{1}{(2k-1)^2},
\qquad \psi_1\left(\tfrac12\right) = \frac{\pi^2}{2},
\qquad \psi_1\left(\tfrac32\right) = \frac{\pi^2}{2} - 4 .
</math>
The trigamma function has other special values such as:
: <math>
\psi_1\left(\tfrac14\right) = \pi^2 + 8G
</math>
where represents Catalan's constant.
There are no roots on the real axis of , but there exist infinitely many pairs of roots for . Each such pair of roots approaches quickly and their imaginary part increases slowly logarithmic with . For example, and are the first two roots with .
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,
:<math>
\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).
</math>
Appearance
The trigamma function appears in this sum formula:
</references>
References
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. . See section §6.4
- Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource