# real variables

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## 1—10 of 498 matching pages

##### 1: 32.1 Special Notation
 $m,n$ integers. real variable. complex variable. …
##### 2: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
##### 3: 4.32 Inequalities
4.32.3 $|\cosh x-\cosh y|\geq|x-y|\sqrt{\sinh x\sinh y},$ $x>0$, $y>0$,
4.32.4 $\operatorname{arctan}x\leq\tfrac{1}{2}\pi\tanh x,$ $x\geq 0$.
##### 6: 4.15 Graphics
4.15.1 $\cos\left(x+iy\right)=\sin\left(x+\tfrac{1}{2}\pi+iy\right),$
4.15.2 $\cot\left(x+iy\right)=-\tan\left(x+\tfrac{1}{2}\pi+iy\right),$
4.15.3 $\sec\left(x+iy\right)=\csc\left(x+\tfrac{1}{2}\pi+iy\right),$
##### 7: 28.30 Expansions in Series of Eigenfunctions
###### §28.30(i) RealVariable
28.30.2 $\frac{1}{2\pi}\int_{0}^{2\pi}w_{m}(x)w_{n}(x)\,\mathrm{d}x=\delta_{m,n}.$
28.30.3 $f(x)=\sum_{m=0}^{\infty}f_{m}w_{m}(x),$
28.30.4 $f_{m}=\frac{1}{2\pi}\int_{0}^{2\pi}f(x)w_{m}(x)\,\mathrm{d}x.$
##### 9: 12.13 Sums
12.13.1 $U\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac% {(-y)^{m}}{m!}U\left(a-m,x\right),$
12.13.2 $U\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U\left(a+m,x\right),$
12.13.3 $V\left(a,x+y\right)=e^{\frac{1}{2}xy+\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{a-\tfrac{1}{2}}{m}y^{m}V\left(a-m,x\right),$
12.13.4 $V\left(a,x+y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \frac{y^{m}}{m!}V\left(a+m,x\right).$
12.13.5 $U\left(a,x\cos t+y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m+a,x\right)U\left(-m-\tfrac{1}{2},y% \right),$ $\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi$.
##### 10: 6.1 Special Notation
 $x$ real variable. …