# on finite point sets

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## 1—10 of 27 matching pages

##### 1: 18.2 General Orthogonal Polynomials

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###### Orthogonality on Finite Point Sets

►Let $X$ be a finite set of distinct points on $\mathbb{R}$, or a countable infinite set of distinct points on $\mathbb{R}$, and ${w}_{x}$, $x\in X$, be a set of positive constants. …when $X$ is a finite set of $N+1$ distinct points. …##### 2: 23.20 Mathematical Applications

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►Let $T$ denote the set of points on $C$ that are of finite order (that is, those points
$P$ for which there exists a positive integer $n$ with $nP=o$), and let $I,K$ be the sets of points with integer and rational coordinates, respectively.
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##### 3: 1.9 Calculus of a Complex Variable

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►A

*domain*$D$, say, is an open set in $\u2102$ 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. …##### 4: 36.12 Uniform Approximation of Integrals

###### §36.12 Uniform Approximation of Integrals

►###### §36.12(i) General Theory for Cuspoids

… ►Also, $f$ is real analytic, and ${\partial}^{K+2}f/{\partial u}^{K+2}>0$ for all $\mathbf{y}$ such that all $K+1$ critical points coincide. … ►In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … ►Also, ${\mathrm{\Delta}}^{1/4}/\sqrt{{f}_{+}^{\prime \prime}}$ and ${\mathrm{\Delta}}^{1/4}/\sqrt{-{f}_{-}^{\prime \prime}}$ are chosen to be positive real when $y$ is such that both critical points are real, and by analytic continuation otherwise. …##### 5: 2.1 Definitions and Elementary Properties

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►Let $\mathbf{X}$ be a point set with a limit point
$c$.
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►If $c$ is a finite limit point of $\mathbf{X}$, then
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►Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$, and for each nonnegative integer $n$
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►Similarly for finite limit point
$c$ in place of $\mathrm{\infty}$.
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►where $c$ is a finite, or infinite, limit point of $\mathbf{X}$.
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##### 6: 21.7 Riemann Surfaces

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►Consider the set of points in ${\u2102}^{2}$ that satisfy the equation
…This compact curve may have singular points, that is, points at which the gradient of $\stackrel{~}{P}$ vanishes.
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►On this surface, we choose $2g$

*cycles*(that is, closed oriented curves, each with at most a finite number of singular points) ${a}_{j}$, ${b}_{j}$, $j=1,2,\mathrm{\dots},g$, such that their*intersection indices*satisfy … ►The zeros ${\lambda}_{j}$, $j=1,2,\mathrm{\dots},2g+1$ of $Q(\lambda )$ specify the finite branch points ${P}_{j}$, that is, points at which ${\mu}_{j}=0$, on the Riemann surface. Denote the set of all branch points by $B=\{{P}_{1},{P}_{2},\mathrm{\dots},{P}_{2g+1},{P}_{\mathrm{\infty}}\}$. …##### 7: 1.10 Functions of a Complex Variable

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►If the poles are infinite in number, then the point at infinity is called an

*essential singularity*: it is the limit point of the poles. … ►A*cut domain*is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed. … ► … ►Alternatively, take ${z}_{0}$ to be any point in $D$ and set $F({z}_{0})={\mathrm{e}}^{\alpha \mathrm{ln}\left(1-{z}_{0}\right)}{\mathrm{e}}^{\beta \mathrm{ln}\left(1+{z}_{0}\right)}$ where the logarithms assume their principal values. … ►(The integer $k$ may be greater than one to allow for a finite number of zero factors.) …##### 8: 1.4 Calculus of One Variable

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►where the supremum is over all sets of points
$$ in the

*closure*of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …##### 9: 36.15 Methods of Computation

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►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7.
Close to the bifurcation set but far from $\mathbf{x}=\mathrm{\U0001d7ce}$, the uniform asymptotic approximations of §36.12 can be used.
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►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi}$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\mathrm{exp}\left(\mathrm{i}\mathrm{\Phi}\right)$.
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###### §36.15(iv) Integration along Finite Contour

►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi}$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …##### 10: 1.5 Calculus of Two or More Variables

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►A function is

*continuous on a point set*$D$ if it is continuous at all points of $D$. … … ►