open disks around infinity

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1: 1.9 Calculus of a Complex Variable

… ►It is single-valued on $ℂ\setminus \{0\}$, except on the interval $(-\mathrm{\infty },0)$ where it is discontinuous and two-valued. … ►

Continuity

… ►at $(x,y)$. … ►A system of open disks around infinity is given by … ►Suppose ${\sum }_{n=0}^{\mathrm{\infty }}{f}_n(t)$ 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 $\sigma (j)\ne k$. The number of derangements of $n$ is the number of permutations with forbidden positions $B=\{(1,1),(2,2),\mathrm{\dots },(n,n)\}$. … ►For $(j,k)\in B$, $B\setminus [j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\mathrm{\dots },n$. $B\setminus (j,k)$ denotes $B$ with the element $(j,k)$ removed. … ►Let . …

3: 14.27 Zeros

… ► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ has no zeros in the interval $(-\mathrm{\infty },-1)$. …

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 $\partial F/\partial y\ne 0$ 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 $+\mathrm{\infty }$, and $f(x,y)$, $\partial f/\partial x$ 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 …

5: 3.5 Quadrature

… ►for some $\xi \in (a,b)$. … ►Or if the set $x_1,x_2,\mathrm{\dots },x_n$ 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 $(a,b)$. … ►

§3.5(x) Cubature Formulas

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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 , , and $r\in (r_1,r_2)$. … ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ 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 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 $(1,1)$. … ►

26.6.6 $$\sum _{n=0}^{\mathrm{\infty }}D(n,n)x^n=\frac{1}{\sqrt{1-6x+x^2}},$$

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Symbols:

$n$: nonnegative integer, $D(m,n)$: Delannoy number and $x$: small

Permalink:

http://dlmf.nist.gov/26.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(ii), §26.6 and Ch.26

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8: 14.16 Zeros

… ►where $m$, $n\in \mathbb{Z}$ and ${\delta }_{\mu }$, ${\delta }_{\nu }\in (0,1)$. … ►The zeros of ${𝖰}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ interlace those of ${𝖯}_{\nu }^{\mu }\left(x\right)$. … ► $P_{\nu }^{\mu }\left(x\right)$ has exactly one zero in the interval $(1,\mathrm{\infty })$ if either of the following sets of conditions holds: … ►For all other values of $\mu$ and $\nu$ (with $\nu \ge -\frac{1}{2}$) $P_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$. ► ${𝑸}_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$ when $\nu >-1$, and at most one zero in the interval $(1,\mathrm{\infty })$ when .

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 $t\in (1,\mathrm{\infty })$. … ►

4.37.19 $$\mathrm{arccosh}z=\ln \left(\pm {(z^2-1)}^{1/2}+z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },1)$,

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction and $z$: complex variable

Referenced by:

§4.37(iv), §4.37(iii), §4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

… ►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 $(-\mathrm{\infty },1]$. … ►On the part of the cut from $-\mathrm{\infty }$ to $-1$ … ►

4.37.24 $$\mathrm{arctanh}z=\frac{1}{2}\ln \left(\frac{1+z}{1-z}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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10: 4.3 Graphics

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Figure 4.3.1: $\ln x$ and ${\mathrm{e}}^x$. Parallel tangent lines at $(1,0)$ and $(0,1)$ make evident the mirror symmetry across the line $y=x$, demonstrating the inverse relationship between the two functions. Magnify

… ►In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty })$. …