# on a point set

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## 1—10 of 112 matching pages

##### 1: 2.1 Definitions and Elementary Properties

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βΊLet $\mathrm{\pi \x9d\x90\x97}$ be a point set with a limit point
$c$.
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βΊ

2.1.16
$$f\beta \x81\u2018(x)\sim {a}_{0}+{a}_{1}\beta \x81\u2019(x-c)+{a}_{2}\beta \x81\u2019{(x-c)}^{2}+\mathrm{\beta \x8b\u2015},$$
$x\to c$ in $\mathrm{\pi \x9d\x90\x97}$,

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βΊIf the set
$\mathrm{\pi \x9d\x90\x97}$ in §2.1(iii) is a closed sector $\mathrm{\Xi \pm}\le \mathrm{ph}\beta \x81\u2018x\le \mathrm{\Xi \xb2}$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}\beta \x81\u2018x\in [\mathrm{\Xi \pm},\mathrm{\Xi \xb2}]$ as $|x|\to \mathrm{\infty}$.
…Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathrm{\pi \x9d\x90\x94}$, and for each nonnegative integer $n$
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##### 2: 1.5 Calculus of Two or More Variables

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βΊA function is

*continuous on a point set*$D$ if it is continuous at all points of $D$. … … βΊFor $f\beta \x81\u2018(x,y)$ defined on a point set $D$ contained in a rectangle $R$, let … βΊwhere $D$ is the image of ${D}^{\ast}$ under a mapping $(u,v)\to (x\beta \x81\u2018(u,v),y\beta \x81\u2018(u,v))$ which is one-to-one except perhaps for a set of points of area zero. …Again the mapping is one-to-one except perhaps for a set of points of volume zero. …##### 3: 1.6 Vectors and Vector-Valued Functions

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βΊand $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary.
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βΊSuppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set
$V$, and $S$ is oriented by its normal being outwards from $V$.
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##### 4: 1.9 Calculus of a Complex Variable

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βΊAn

*open set*in $\mathrm{\beta \x84\x82}$ is one in which each point has a neighborhood that is contained in the set. βΊA point ${z}_{0}$ is a*limit point*(*limiting point*or*accumulation point*) of a set of points $S$ in $\mathrm{\beta \x84\x82}$ (or $\mathrm{\beta \x84\x82}\cup \mathrm{\infty}$) if every neighborhood of ${z}_{0}$ contains a point of $S$ distinct from ${z}_{0}$. … βΊA*domain*$D$, say, is an open set in $\mathrm{\beta \x84\x82}$ that is*connected*, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … βΊConversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f\beta \x81\u2018(z)$ is differentiable at $z=x+\mathrm{i}\beta \x81\u2019y$. … βΊ
1.9.49
$$R=\underset{n\to \mathrm{\infty}}{lim\; inf}{\left|{a}_{n}\right|}^{-1/n}.$$

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##### 5: 2.8 Differential Equations with a Parameter

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βΊin which $u$ is a real or complex parameter, and asymptotic solutions are needed for large $|u|$ that are uniform with respect to $z$ in a point set
$\mathrm{\pi \x9d\x90\x83}$ in $\mathrm{\beta \x84\x9d}$ or $\mathrm{\beta \x84\x82}$.
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βΊCorresponding to each positive integer $n$ there are solutions ${W}_{n,j}\beta \x81\u2018(u,\mathrm{\Xi \u038e})$, $j=1,2$, that depend on arbitrarily chosen reference points
${\mathrm{\Xi \pm}}_{j}$, are ${C}^{\mathrm{\infty}}$ or analytic on $\mathrm{\pi \x9d\x9a\xab}$, and as $u\to \mathrm{\infty}$
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##### 6: 21.3 Symmetry and Quasi-Periodicity

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βΊThe set of points
${\mathrm{\pi \x9d\x90\xa6}}_{1}+\mathrm{\pi \x9d\x9b\x80}\beta \x81\u2019{\mathrm{\pi \x9d\x90\xa6}}_{2}$ form a
$g$-dimensional lattice, the

*period lattice*of the Riemann theta function. …##### 7: 18.2 General Orthogonal Polynomials

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βΊLet $X$ be a finite set of distinct points on $\mathrm{\beta \x84\x9d}$, or a countable infinite set of distinct points on $\mathrm{\beta \x84\x9d}$, and ${w}_{x}$, $x\in X$, be a set of positive constants.
…when $X$ is a finite set of $N+1$ distinct points.
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βΊIf the polynomials ${p}_{n}\beta \x81\u2018(x)$ ($n=0,1,\mathrm{\dots},N$) are orthogonal on a finite set
$X$ of $N+1$ distinct points as in (18.2.3), then the polynomial ${p}_{N+1}\beta \x81\u2018(x)$ of degree $N+1$, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on $X$.
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##### 8: 26.13 Permutations: Cycle Notation

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βΊAn element of ${\mathrm{\pi \x9d\x94\x96}}_{n}$ with ${a}_{1}$ fixed points, ${a}_{2}$ cycles of length $2,\mathrm{\dots},{a}_{n}$ cycles of length $n$, where $n={a}_{1}+2\beta \x81\u2019{a}_{2}+\mathrm{\beta \x8b\u2015}+n\beta \x81\u2019{a}_{n}$, is said to have

*cycle type*$\left({a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}\right)$. …##### 9: 22.19 Physical Applications

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βΊAs $a\to \sqrt{1/\mathrm{\Xi \xb2}}$ from below the period diverges since $a=\pm \sqrt{1/\mathrm{\Xi \xb2}}$ are points of unstable equilibrium.
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βΊFor an initial displacement with $$, bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\mathrm{\Xi \xb2}}$.
…As $a\to \sqrt{2/\mathrm{\Xi \xb2}}$ from below the period diverges since $x=0$ is a point of unstable equlilibrium.
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##### 10: 3.11 Approximation Techniques

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βΊHere ${x}_{j}$, $j=1,2,\mathrm{\dots},J$, is a given set of distinct real points and $J\ge n+1$.
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