# complex variables

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## 11—20 of 498 matching pages

##### 11: 4.28 Definitions and Periodicity
4.28.2 $\cosh z=\frac{e^{z}+e^{-z}}{2},$
4.28.5 $\operatorname{csch}z=\frac{1}{\sinh z},$
##### 12: 4.7 Derivatives and Differential Equations
4.7.6 $w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}.$
4.7.10 $\frac{\mathrm{d}}{\mathrm{d}z}z^{a}=az^{a-1},$
4.7.14 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=aw,$ $a\neq 0$,
##### 13: 5.18 $q$-Gamma and $q$-Beta Functions
5.18.1 $\left(a;q\right)_{n}=\prod_{k=0}^{n-1}(1-aq^{k}),$ $n=0,1,2,\dots$,
5.18.3 $\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k}).$
5.18.10 $\lim_{q\to 1-}\Gamma_{q}\left(z\right)=\Gamma\left(z\right).$
5.18.11 $\mathrm{B}_{q}\left(a,b\right)=\frac{\Gamma_{q}\left(a\right)\Gamma_{q}\left(b% \right)}{\Gamma_{q}\left(a+b\right)}.$
##### 14: 28.19 Expansions in Series of $\mathrm{me}_{\nu+2n}$ Functions
28.19.1 $f(z+\pi)=e^{\mathrm{i}\nu\pi}f(z).$
28.19.4 $e^{\mathrm{i}\nu z}=\sum_{n=-\infty}^{\infty}c^{\nu+2n}_{-2n}(q)\mathrm{me}_{% \nu+2n}\left(z,q\right),$
##### 15: 7.4 Symmetry
7.4.1 $\operatorname{erf}\left(-z\right)=-\operatorname{erf}\left(z\right),$
7.4.2 $\operatorname{erfc}\left(-z\right)=2-\operatorname{erfc}\left(z\right),$
7.4.4 $F\left(-z\right)=-F\left(z\right).$
##### 16: 4.22 Infinite Products and Partial Fractions
4.22.5 $\csc z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}.$
##### 17: 4.36 Infinite Products and Partial Fractions
4.36.4 ${\operatorname{csch}}^{2}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},$
##### 18: 12.8 Recurrence Relations and Derivatives
12.8.1 $zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0,$
12.8.2 $U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2})U\left(a+1,z% \right)=0,$
12.8.3 $U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)=0,$
12.8.4 $2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+1,z\right)=0.$
12.8.5 $zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-1,z\right)=0,$
##### 19: 5.12 Beta Function
5.12.1 $\mathrm{B}\left(a,b\right)=\int_{0}^{1}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{% \Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}.$
5.12.4 $\int_{0}^{1}\frac{t^{a-1}(1-t)^{b-1}}{(t+z)^{a+b}}\mathrm{d}t=\mathrm{B}\left(% a,b\right)(1+z)^{-a}z^{-b},$ $|\operatorname{ph}z|<\pi$.
5.12.8 ${\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\mathrm{d}t}{(w+it)^{a}(z-it)^{b}}% =\frac{(w+z)^{1-a-b}}{(a+b-1)\mathrm{B}\left(a,b\right)}},$ $\Re\left(a+b\right)>1$, $\Re w>0$, $\Re z>0$.
5.12.12 $\int_{P}^{(1+,0+,1-,0-)}t^{a-1}(1-t)^{b-1}\mathrm{d}t=-4e^{\pi i(a+b)}\sin% \left(\pi a\right)\sin\left(\pi b\right)\mathrm{B}\left(a,b\right),$
##### 20: 4.33 Maclaurin Series and Laurent Series
4.33.3 $\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}z^{7}+\cdots+\frac{2% ^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$.