# open disks around infinity

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##### 1: 1.9 Calculus of a Complex Variable
It is single-valued on $\mathbb{C}\setminus\{0\}$, except on the interval $(-\infty,0)$ where it is discontinuous and two-valued. …
###### Continuity
at $(x,y)$. … A system of open disks around infinity is given by … Suppose $\sum^{\infty}_{n=0}f_{n}(t)$ converges uniformly in any compact interval in $(a,b)$, and at least one of the following two conditions is satisfied: …
##### 2: 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. …
##### 3: 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 $(-\infty,-1)$. …
##### 4: 1.5 Calculus of Two or More Variables
A function $f(x,y)$ is continuous at a point $(a,b)$ if … If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\ifrac{\partial F}{\partial y}\not=0$ at $(a,b)$, then in a neighborhood of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and $g^{\prime}(x)=-F_{x}/F_{y}$. … $f(x,y)$ has a local minimum (maximum) at $(a,b)$ if … Suppose that $a,b,c$ are finite, $d$ is finite or $+\infty$, and $f(x,y)$, $\ifrac{\partial f}{\partial x}$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times[c,d)$. … Moreover, if $a,b,c,d$ are finite or infinite constants and $f(x,y)$ is piecewise continuous on the set $(a,b)\times(c,d)$, then …
for some $\xi\in(a,b)$. … Or if the set $x_{1},x_{2},\dots,x_{n}$ lies in the open interval $(a,b)$, then the quadrature rule is said to be open. … Examples of open rules are the Gauss formulas (§3.5(v)), the midpoint rule, and Fejér’s quadrature rule. … and $\xi$ is some point in $(a,b)$. …
##### 6: 1.10 Functions of a Complex Variable
We write $(f_{1},D_{1})$, $(f_{2},D_{2})$ to signify this continuation. … Suppose $f(z)$ is analytic in the annulus $r_{1}<\left|z-z_{0}\right|, $0\leq r_{1}, and $r\in(r_{1},r_{2})$. … If $D=\mathbb{C}\setminus(-\infty,0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta}$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta/2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta/2}$, with $-\pi<\theta<\pi$ in both cases. … (Thus if $z_{0}$ is in the interval $(-1,1)$, then the logarithms are real.) … Let $F(x,z)$ have a converging power series expansion of the form …
##### 7: 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 $(1,1)$. …
26.6.9 $\sum_{n=0}^{\infty}r(n)x^{n}=\frac{1-x-\sqrt{1-6x+x^{2}}}{2x}.$
##### 8: 14.16 Zeros
where $m$, $n\in\mathbb{Z}$ and $\delta_{\mu}$, $\delta_{\nu}\in(0,1)$. … The zeros of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ in the interval $(-1,1)$ interlace those of $\mathsf{P}^{\mu}_{\nu}\left(x\right)$. … $P^{\mu}_{\nu}\left(x\right)$ has exactly one zero in the interval $(1,\infty)$ if either of the following sets of conditions holds: … For all other values of $\mu$ and $\nu$ (with $\nu\geq-\frac{1}{2}$) $P^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$. $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ has no zeros in the interval $(1,\infty)$ when $\nu>-1$, and at most one zero in the interval $(1,\infty)$ when $\nu<-1$.
##### 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 $(-\infty,1]$. … On the part of the cut from $-\infty$ to $-1$
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: 18.40 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 $i$. …
18.40.9 $x(t,N)=\cfrac{x_{1,N}}{1+\cfrac{a_{1}(t-1)}{1+\cfrac{a_{2}(t-2)}{1+\cdots}}}% \frac{a_{N-1}(t-(N-1))}{1},$ $t\in(0,\infty)$,
The PWCF $x(t,N)$ is a minimally oscillatory algebraic interpolation of the abscissas $x_{i,N},i=1,2,\dots,N$. … 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). …