Editing Digamma function (section)

==Special values==
The digamma function has values in closed form for rational numbers, as a result of [[#Gauss's digamma theorem|Gauss's digamma theorem]]. Some are listed below:

:<math>\begin{align}
\psi(1) &= -\gamma \\ 
\psi\left(\tfrac{1}{2}\right) &= -2\ln{2} - \gamma \\
\psi\left(\tfrac{1}{3}\right) &= -\frac{\pi}{2\sqrt{3}} -\frac{3\ln{3}}{2} - \gamma \\ 
\psi\left(\tfrac{1}{4}\right) &= -\frac{\pi}{2} - 3\ln{2} - \gamma \\
\psi\left(\tfrac{1}{6}\right) &= -\frac{\pi\sqrt{3}}{2} -2\ln{2} -\frac{3\ln{3}}{2} - \gamma \\
\psi\left(\tfrac{1}{8}\right) &= -\frac{\pi}{2} - 4\ln{2} - \frac {\pi + \ln \left (\sqrt{2} + 1 \right ) - \ln \left (\sqrt{2} - 1 \right ) }{\sqrt{2}} - \gamma.
\end{align}</math>

Moreover, by taking the logarithmic derivative of <math>|\Gamma (bi)|^2</math> or <math>|\Gamma (\tfrac{1}{2}+bi)|^2</math> where <math>b</math> is real-valued, it can easily be deduced that

:<math>\operatorname{Im} \psi(bi) = \frac{1}{2b}+\frac{\pi}{2}\coth (\pi b),</math>
:<math>\operatorname{Im} \psi(\tfrac{1}{2}+bi) = \frac{\pi}{2}\tanh (\pi b).</math>

Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the [[imaginary unit]] the numerical approximation
:<math>\operatorname{Re} \psi(i) = -\gamma-\sum_{n=0}^\infty\frac{n-1}{n^3+n^2+n+1} \approx 0.09465.</math>