on finite point sets

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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 $ℝ$, or a countable infinite set of distinct points on $ℝ$, 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

… ►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. …

3: 1.9 Calculus of a Complex Variable

… ►A domain $D$, say, is an open set in $ℂ$ 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

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§36.12(i) General Theory for Cuspoids

… ►Also, $f$ is real analytic, and ${\partial }^{K+2}f/{\partial u}^{K+2}>0$ for all $𝐲$ 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

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

6: 21.7 Riemann Surfaces

… ►Consider the set of points in ${ℂ}^2$ that satisfy the equation …This compact curve may have singular points, that is, points at which the gradient of $\tilde{P}$ vanishes. … ►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

… ►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 $F(z_0)={\mathrm{e}}^{\alpha \ln \left(1-z_0\right)}{\mathrm{e}}^{\beta \ln \left(1+z_0\right)}$ 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

… ►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

… ►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 $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. … ►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 $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. … ►

§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

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

Finite Integrals

… ►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 … ►Again the mapping is one-to-one except perhaps for a set of points of volume zero. …