# psi function

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##### 1: 5.2 Definitions
###### Euler’s Integral
5.2.1 $\Gamma\left(z\right)=\int_{0}^{\infty}e^{-t}t^{z-1}\mathrm{d}t,$ $\Re z>0$.
5.2.2 $\psi\left(z\right)=\Gamma'\left(z\right)/\Gamma\left(z\right),$ $z\neq 0,-1,-2,\dots$.
##### 2: 5.1 Special Notation
 $j,m,n$ nonnegative integers. …
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: 5.21 Methods of Computation
For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). Similarly for $\ln\Gamma\left(z\right)$, $\psi\left(z\right)$, and the polygamma functions. …
##### 4: 5.23 Approximations
###### §5.23(iii) Approximations in the Complex Plane
See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\Gamma\left(z\right)$. …
##### 5: 5.16 Sums
5.16.2 $\sum_{k=1}^{\infty}\frac{1}{k}\psi'\left(k+1\right)=\zeta\left(3\right)=-\frac% {1}{2}\psi''\left(1\right).$
For further sums involving the psi function see Hansen (1975, pp. 360–367). …
##### 7: 5.3 Graphics Figure 5.3.3: ψ ⁡ ( x ) . Magnify Figure 5.3.6: | ψ ⁡ ( x + i ⁢ y ) | . Magnify 3D Help
##### 8: 5.5 Functional Relations
5.5.1 $\Gamma\left(z+1\right)=z\Gamma\left(z\right),$
5.5.2 $\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}.$
5.5.3 $\Gamma\left(z\right)\Gamma\left(1-z\right)=\pi/\sin\left(\pi z\right),$ $z\neq 0,\pm 1,\dots$,
5.5.8 $\psi\left(2z\right)=\tfrac{1}{2}\left(\psi\left(z\right)+\psi\left(z+\tfrac{1}% {2}\right)\right)+\ln 2,$
##### 9: 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(ii) Other Series
5.7.7 $\psi\left(\frac{z+1}{2}\right)-\psi\left(\frac{z}{2}\right)=2\sum_{k=0}^{% \infty}\frac{(-1)^{k}}{k+z}.$
##### 10: 5.4 Special Values and Extrema
###### §5.4(ii) PsiFunction
5.4.19 $\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{\pi}{2}\cot\left(\frac{\pi p}% {q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(\frac{2\pi kp}{q}\right)\ln% \left(2-2\cos\left(\frac{2\pi k}{q}\right)\right).$
###### §5.4(iii) Extrema
As $n\to\infty$, …