# psi function

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## 11—20 of 100 matching pages

##### 11: 5.15 Polygamma Functions
The functions $\psi^{(n)}\left(z\right)$, $n=1,2,\dots$, are called the 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. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. …
5.15.1 $\psi'\left(z\right)=\sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}},$ $z\neq 0,-1,-2,\dots$,
##### 13: 5.19 Mathematical Applications
###### §5.19(i) Summation of Rational Functions
5.19.3 $S=\psi\left(\tfrac{1}{2}\right)-2\psi\left(\tfrac{2}{3}\right)-\gamma=3\ln 3-2% \ln 2-\tfrac{1}{3}\pi\sqrt{3}.$
##### 14: 25.16 Mathematical Applications
In studying the distribution of primes $p\leq x$, Chebyshev (1851) introduced a function $\psi\left(x\right)$ (not to be confused with the digamma function used elsewhere in this chapter), given by
25.16.1 $\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p,$
25.16.2 $\psi\left(x\right)=x-\frac{\zeta'\left(0\right)}{\zeta\left(0\right)}-\sum_{% \rho}\frac{x^{\rho}}{\rho}+o\left(1\right),$ $x\to\infty$,
25.16.3 $\psi\left(x\right)=x+o\left(x\right),$ $x\to\infty$.
25.16.4 $\psi\left(x\right)=x+O\left(x^{\frac{1}{2}+\epsilon}\right),$ $x\to\infty$,
##### 15: 5.6 Inequalities
###### Kershaw’s Inequality
5.6.5 $\exp\left((1-s)\psi\left(x+s^{1/2}\right)\right)\leq\frac{\Gamma\left(x+1% \right)}{\Gamma\left(x+s\right)}\leq\exp\left((1-s)\psi\left(x+\tfrac{1}{2}(s+% 1)\right)\right),$ $0.
##### 16: 36.6 Scaling Relations
$\Psi_{K}(\mathbf{x};k)=k^{\beta_{K}}\Psi_{K}\left(\mathbf{y}(k)\right),$
$\Psi^{(\mathrm{U})}(\mathbf{x};k)=k^{\beta^{(\mathrm{U})}}\Psi^{(\mathrm{U})}% \left(\mathbf{y}^{(\mathrm{U})}(k)\right),$
##### 18: 36.2 Catastrophes and Canonical Integrals
36.2.5 $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\exp\left(i\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)% \right)\mathrm{d}s\mathrm{d}t,$ $\mathrm{U}=\mathrm{E},\mathrm{H}$.
36.2.11 $\Psi^{(\mathrm{U})}(\mathbf{x};k)=k\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\exp\left(ik\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)% \mathrm{d}s\mathrm{d}t,$ $\mathrm{U=E,H}$; $k>0$.
$\Psi_{1}$ is related to the Airy function9.2):
36.2.15 $\Psi_{K}\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\Gamma\left(\frac{1}{K+2}% \right)\*\begin{cases}\exp\left(i\dfrac{\pi}{2(K+2)}\right),&K\text{ even,}\\ \cos\left(\dfrac{\pi}{2(K+2)}\right),&K\text{ odd}.\end{cases}$
##### 19: 17.1 Special Notation
The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, and the $q$-analogs of the Appell functions $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)$, $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)$, $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)$, and $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)$. …
##### 20: 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 $\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}_{\nu}\left(x\right)\right|_% {\mu=0}=-\tfrac{1}{4}\pi^{2}\mathsf{P}_{\nu}\left(x\right)+\left(\psi\left(-% \nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{Q}_{\nu}\left(x\right).$