# digamma function

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##### 1: 5.16 Sums
5.16.2 $\sum_{k=1}^{\infty}\frac{1}{k}\psi'\left(k+1\right)=\zeta\left(3\right)=-\frac% {1}{2}\psi''\left(1\right).$
##### 2: 5.1 Special Notation
The main functions treated in this chapter are the gamma function $\Gamma\left(z\right)$, the psi function (or digamma function) $\psi\left(z\right)$, the beta function $\mathrm{B}\left(a,b\right)$, and the $q$-gamma function $\Gamma_{q}\left(z\right)$. … Alternative notations for the psi function are: $\Psi(z-1)$ (Gauss) Jahnke and Emde (1945); $\Psi(z)$ Davis (1933); $\mathsf{F}(z-1)$ Pairman (1919).
##### 3: 5.15 Polygamma Functions
In particular, $\psi'\left(z\right)$ is the trigamma function; $\psi''$, $\psi^{(3)}$, $\psi^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. …
5.15.1 $\psi'\left(z\right)=\sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}},$ $z\neq 0,-1,-2,\dots$,
5.15.6 ${\psi}^{(n)}\left(1-z\right)+(-1)^{n-1}{\psi}^{(n)}\left(z\right)=(-1)^{n}\pi% \frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cot\left(\pi z\right),$
##### 4: 5.7 Series Expansions
5.7.5 $\psi\left(1+z\right)=\frac{1}{2z}-\frac{\pi}{2}\cot\left(\pi z\right)+\frac{1}% {z^{2}-1}+1-\gamma-\sum_{k=1}^{\infty}(\zeta\left(2k+1\right)-1)z^{2k},$ $|z|<2$, $z\neq 0,\pm 1$.
5.7.6 $\psi\left(z\right)=-\gamma-\frac{1}{z}+\sum_{k=1}^{\infty}\frac{z}{k(k+z)}=-% \gamma+\sum_{k=0}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+z}\right),$
5.7.7 $\psi\left(\frac{z+1}{2}\right)-\psi\left(\frac{z}{2}\right)=2\sum_{k=0}^{% \infty}\frac{(-1)^{k}}{k+z}.$
##### 5: 5.4 Special Values and Extrema
5.4.15 $\psi\left(n+\tfrac{1}{2}\right)=-\gamma-2\ln 2+2\left(1+\tfrac{1}{3}+\dots+% \tfrac{1}{2n-1}\right),$ $n=1,2,\dots$.
5.4.16 $\Im\psi\left(iy\right)=\frac{1}{2y}+\frac{\pi}{2}\coth\left(\pi y\right),$
5.4.17 $\Im\psi\left(\tfrac{1}{2}+iy\right)=\frac{\pi}{2}\tanh\left(\pi y\right),$
5.4.19 $\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{\pi}{2}\cot\left(\frac{\pi p}% {q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(\frac{2\pi kp}{q}\right)\ln% \left(2-2\cos\left(\frac{2\pi k}{q}\right)\right).$
##### 6: 25.1 Special Notation
 $k,m,n$ nonnegative integers. … digamma function $\Gamma'\left(x\right)/\Gamma\left(x\right)$ except in §25.16. See §5.2(i). …
##### 7: 5.5 Functional Relations
5.5.2 $\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}.$
5.5.4 $\psi\left(z\right)-\psi\left(1-z\right)=-\pi/\tan\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$.
5.5.8 $\psi\left(2z\right)=\tfrac{1}{2}\left(\psi\left(z\right)+\psi\left(z+\tfrac{1}% {2}\right)\right)+\ln 2,$
##### 8: 5.3 Graphics
###### §5.3 Graphics
In the graphics shown in this subsection, both the height and color correspond to the absolute value of the function. …
##### 9: 14.11 Derivatives with Respect to Degree or Order
14.11.3 $\mathsf{A}_{\nu}^{\mu}(x)=\sin\left(\nu\pi\right)\left(\frac{1+x}{1-x}\right)^% {\mu/2}\*\sum_{k=0}^{\infty}\frac{\left(\frac{1}{2}-\frac{1}{2}x\right)^{k}% \Gamma\left(k-\nu\right)\Gamma\left(k+\nu+1\right)}{k!\Gamma\left(k-\mu+1% \right)}\*\left(\psi\left(k+\nu+1\right)-\psi\left(k-\nu\right)\right).$
14.11.4 $\left.\frac{\partial}{\partial\mu}\mathsf{P}^{\mu}_{\nu}\left(x\right)\right|_% {\mu=0}=\left(\psi\left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{P% }_{\nu}\left(x\right)+\mathsf{Q}_{\nu}\left(x\right),$
14.11.5 $\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right|_% {\mu=0}=-\tfrac{1}{4}\pi^{2}\mathsf{P}_{\nu}\left(x\right)+\left(\psi\left(-% \nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{Q}_{\nu}\left(x\right).$
##### 10: 5.2 Definitions
5.2.2 $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$ $z\neq 0,-1,-2,\dots$.