# closed

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## 1—10 of 139 matching pages

##### 1: 7.23 Tables
• Abramowitz and Stegun (1964, Chapter 7) includes $\operatorname{erf}x$, $(2/\sqrt{\pi})e^{-x^{2}}$, $x\in[0,2]$, 10D; $(2/\sqrt{\pi})e^{-x^{2}}$, $x\in[2,10]$, 8S; $xe^{x^{2}}\operatorname{erfc}x$, $x^{-2}\in[0,0.25]$, 7D; $2^{n}\Gamma\left(\frac{1}{2}n+1\right)\mathop{\mathrm{i}^{n}\mathrm{erfc}}% \left(x\right)$, $n=1(1)6,10,11$, $x\in[0,5]$, 6S; $F\left(x\right)$, $x\in[0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in[0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in[0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in[0,1]$, $x^{-1}\in[0,1]$, 15D.

• Finn and Mugglestone (1965) includes the Voigt function $H\left(a,u\right)$, $u\in[0,22]$, $a\in[0,1]$, 6S.

• Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$, $x\in[0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\mathrm{d}t$, $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\mathrm{d}t$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

• ##### 2: 29.9 Stability
If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\leq a^{0}_{\nu}\left(k^{2}\right)$ or $h$ lies in one of the closed intervals with endpoints $a^{m}_{\nu}\left(k^{2}\right)$ and $b^{m}_{\nu}\left(k^{2}\right)$, $m=1,2,\dots$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\leq a^{0}_{\nu}\left(k^{2}\right)$ or $h\in[b^{m}_{\nu}\left(k^{2}\right),a^{m}_{\nu}\left(k^{2}\right)]$ for some $m=1,2,\dots,\nu$.
##### 3: 22.17 Moduli Outside the Interval [0,1]
###### §22.17 Moduli Outside the Interval [0,1]
Jacobian elliptic functions with real moduli in the intervals $(-\infty,0)$ and $(1,\infty)$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas. … For proofs of these results and further information see Walker (2003).
##### 4: Mathematical Introduction
 $\mathbb{C}$ complex plane (excluding infinity). … $f(z)$ is continuous at all points of a simple closed contour $C$ in $\mathbb{C}$. … closed interval in $\mathbb{R}$, or closed straight-line segment joining $a$ and $b$ in $\mathbb{C}$.
 $(a,b]$ or $[a,b)$ half-closed intervals. …
Mathematically, we scale the height to $h$ lying in the interval $[0,4]$ and the components are computed as follows … Specifically, by scaling the phase angle in $[0,2\pi)$ to $q$ in the interval $[0,4)$, the hue (in degrees) is computed as …
##### 6: 4.37 Inverse Hyperbolic Functions
4.37.17 $\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^{% 1/2}+y\right),$ $y\in[1,\infty)$,
4.37.18 $\operatorname{arcsinh}\left(iy\right)=-\tfrac{1}{2}\pi i\pm\ln\left((y^{2}-1)^% {1/2}-y\right),$ $y\in(-\infty,-1]$,
It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\infty,1]$. …
4.37.22 $\operatorname{arccosh}x=\pm\ln\left(i(1-x^{2})^{1/2}+x\right),$ $x\in(-1,1]$,
4.37.24 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;
##### 7: 5.14 Multidimensional Integrals
5.14.4 $\int_{[0,1]^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}\prod_{k% =1}^{n}t_{k}^{a-1}(1-t_{k})^{b-1}\mathrm{d}t_{k}=\frac{1}{(\Gamma\left(1+c% \right))^{n}}\prod_{k=1}^{m}\frac{a+(n-k)c}{a+b+(2n-k-1)c}\*\prod_{k=1}^{n}% \frac{\Gamma\left(a+(n-k)c\right)\Gamma\left(b+(n-k)c\right)\Gamma\left(1+kc% \right)}{\Gamma\left(a+b+(2n-k-1)c\right)},$
5.14.5 $\int_{[0,\infty)^{n}}t_{1}t_{2}\cdots t_{m}|\Delta(t_{1},\dots,t_{n})|^{2c}% \prod_{k=1}^{n}t_{k}^{a-1}e^{-t_{k}}\mathrm{d}t_{k}=\prod_{k=1}^{m}(a+(n-k)c)% \frac{\prod_{k=1}^{n}\Gamma\left(a+(n-k)c\right)\Gamma\left(1+kc\right)}{(% \Gamma\left(1+c\right))^{n}},$
5.14.7 $\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j $\Re b>-1/n$.
##### 8: 22.18 Mathematical Applications
With $k\in[0,1]$ the mapping $z\to w=\operatorname{sn}\left(z,k\right)$ gives a conformal map of the closed rectangle $[-K,K]\times[0,K^{\prime}]$ onto the half-plane $\Im w\geq 0$, with $0,\pm K,\pm K+iK^{\prime},iK^{\prime}$ mapping to $0,\pm 1,\pm k^{-2},\infty$ respectively. The half-open rectangle $(-K,K)\times[-K^{\prime},K^{\prime}]$ maps onto $\mathbb{C}$ cut along the intervals $(-\infty,-1]$ and $[1,\infty)$. …
##### 9: 7.20 Mathematical Applications Figure 7.20.1: Cornu’s spiral, formed from Fresnel integrals, is defined parametrically by x = C ⁡ ( t ) , y = S ⁡ ( t ) , t ∈ [ 0 , ∞ ) . Magnify
##### 10: 18.24 Hahn Class: Asymptotic Approximations
With $\mu=N/n$ and $x$ fixed, Qiu and Wong (2004) gives an asymptotic expansion for $K_{n}\left(x;p,N\right)$ as $n\to\infty$, that holds uniformly for $\mu\in[1,\infty)$. … Taken together, these expansions are uniformly valid for $-\infty and for $a$ in unbounded intervals—each of which contains $[0,(1-\delta)n]$, where $\delta$ again denotes an arbitrary small positive constant. … These approximations are in terms of Laguerre polynomials and hold uniformly for $\operatorname{ph}\left(x+i\lambda\right)\in[0,\pi]$. …