# on intervals

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## 1—10 of 237 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: 14.27 Zeros
$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P^{\mu}_{\nu}\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$. …
##### 3: 26.15 Permutations: Matrix Notation
If $(j,k)\in B$, then $\sigma(j)\neq k$. The number of derangements of $n$ is the number of permutations with forbidden positions $B=\{(1,1),(2,2),\ldots,(n,n)\}$. … For $(j,k)\in B$, $B\setminus[j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\ldots,n$. $B\setminus(j,k)$ denotes $B$ with the element $(j,k)$ removed. … Let $B=\{(j,j),(j,j+1)\>|\>1\leq j. …
##### 4: 14.16 Zeros
###### §14.16(ii) Interval$-1
The zeros of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ in the interval $(-1,1)$ interlace those of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$. …
###### §14.16(iii) Interval$1
$P^{\mu}_{\nu}\left(x\right)$ has exactly one zero in the interval $(1,\infty)$ if either of the following sets of conditions holds: … $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$ when $\nu>-1$, and at most one zero in the interval $(1,\infty)$ when $\nu<-1$.
##### 5: 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).
##### 6: 1.4 Calculus of One Variable
Suppose $f(x)$ is defined on $[a,b]$. … Then for $f(x)$ continuous on $(a,b)$, … for any $c,d\in(a,b)$, and $t\in[0,1]$. …A similar definition applies to closed intervals $[a,b]$. …
##### 7: 18.40 Methods of Computation
Let $x^{\prime}\in(a,b)$. … Here $x(t,N)$ is an interpolation of the abscissas $x_{i,N},i=1,2,\dots,N$, that is, $x(i,N)=x_{i,N}$, allowing differentiation by $i$. …The PWCF $x(t,N)$ is a minimally oscillatory algebraic interpolation of the abscissas $x_{i,N},i=1,2,\dots,N$. … This is a challenging case as the desired $w^{\mathrm{RCP}}(x)$ on $[-1,1]$ has an essential singularity at $x=-1$. … Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in[-1,1]$ and $(-\infty,\infty)$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …
##### 8: 26.6 Other Lattice Path Numbers
$D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … $r(n)$ is the number of paths from $(0,0)$ to $(n,n)$ that stay on or above the diagonal $y=x$ and are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. …
##### 9: 4.37 Inverse Hyperbolic Functions
In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function $(t^{2}-1)^{1/2}$ assumes its principal value when $t\in(1,\infty)$. …
4.37.19 $\operatorname{arccosh}z=\ln\left(\pm(z^{2}-1)^{1/2}+z\right),$ $z\in\mathbb{C}\setminus(-\infty,1)$,
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 $\operatorname{arctanh}z=\tfrac{1}{2}\ln\left(\frac{1+z}{1-z}\right),$ $z\in\mathbb{C}\setminus(-\infty,-1]\cup[1,\infty)$;
##### 10: 18.16 Zeros
Let $\theta_{n,m}=\theta_{n,m}^{(\alpha,\beta)}$, $m=1,2,\dots,n$, denote the zeros of $P^{(\alpha,\beta)}_{n}\left(\cos\theta\right)$ as function of $\theta$ with …
18.16.2 $\theta_{n,m}^{(-\frac{1}{2},\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{% 1}{2}}\leq\theta_{n,m}^{(\alpha,\beta)}\leq\frac{m\pi}{n+\tfrac{1}{2}}=\theta_% {n,m}^{(\frac{1}{2},-\frac{1}{2})},$ $\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]$,
18.16.3 $\theta_{n,m}^{(-\frac{1}{2},-\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n}\leq% \theta_{n,m}^{(\alpha,\alpha)}\leq\frac{m\pi}{n+1}=\theta_{n,m}^{(\frac{1}{2},% \frac{1}{2})},$ $\alpha\in[-\tfrac{1}{2},\tfrac{1}{2}]$, $m=1,2,\dots,\left\lfloor\frac{1}{2}n\right\rfloor$.
when $\alpha\notin(-\frac{1}{2},\frac{1}{2})$. … All zeros of $H_{n}\left(x\right)$ lie in the open interval $(-\sqrt{2n+1},\sqrt{2n+1})$. …