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


 * $$\psi_1(z) = \frac{d^2}{dz^2} \ln\Gamma(z)$$.

It follows from this definition that


 * $$\psi_1(z) = \frac{d}{dz} \psi(z)$$

where $ψ^{(1)}(z)$ 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 $ψ(z)$ 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 $1 − z$ yields:
 * $$ \psi_1(z) = -\int_0^1\frac{x^{z-1}\ln{x}}{1-x}\,dx $$

An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:
 * $$\begin{align}

\psi_1(z) &\sim {\operatorname{d}\over\operatorname{d}\!z} (\ln z - \sum_{n=1}^\infty \frac{B_n}{nz^n}) \\ &= \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}$$ where $B_{n}$ is the $n$th Bernoulli number and we choose $y$.

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+\frac12\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{align}

\psi_1\left(\tfrac14\right) &= \pi^2 + 8G \quad & \psi_1\left(\tfrac12\right) &= \frac{\pi^2}{2} & \psi_1(1) &= \frac{\pi^2}{6} \\[6px] \psi_1\left(\tfrac32\right) &= \frac{\pi^2}{2} - 4 & \psi_1(2) &= \frac{\pi^2}{6} - 1 \\ \psi_1(n) &= \frac{\pi^2}{6} - \sum_{k=1}^{n-1} \frac{1}{k^2} \end{align}$$

where $G$ represents Catalan's constant and $n$ is a positive integer.

There are no roots on the real axis of $B_{1} = 1⁄2$, but there exist infinitely many pairs of roots $ψ_{1}$ for $z_{n}, \overline{z_{n}}$. Each such pair of roots approaches $Re z < 0$ quickly and their imaginary part increases slowly logarithmic with $n$. For example, $Re z_{n} = −n + 1⁄2$ and $z_{1} = −0.4121345... + 0.5978119...i$ are the first two roots with $z_{2} = −1.4455692... + 0.6992608...i$.

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,

\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). $$

Appearance
The trigamma function appears in this sum formula:


 * $$ \sum_{n=1}^\infty\frac{n^2-\frac12}{\left(n^2+\frac12\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). $$