# on finite point sets

(0.003 seconds)

## 1—10 of 27 matching pages

##### 1: 18.2 General Orthogonal Polynomials
###### Orthogonality on FinitePointSets
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
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 $\mathbb{C}$ 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(i) General Theory for Cuspoids
Also, $f$ is real analytic, and $\ifrac{{\partial}^{K+2}f}{{\partial u}^{K+2}}>0$ 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, $\Delta^{1/4}/\sqrt{f_{+}^{\prime\prime}}$ and $\Delta^{1/4}/\sqrt{-f_{-}^{\prime\prime}}$ 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 $\mathbf{X}$ be a point set with a limit point $c$. … If $c$ is a finite limit point of $\mathbf{X}$, then … Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $\mathbf{U}$, and for each nonnegative integer $n$Similarly for finite limit point $c$ in place of $\infty$. … where $c$ is a finite, or infinite, limit point of $\mathbf{X}$. …
##### 6: 21.7 Riemann Surfaces
Consider the set of points in ${\mathbb{C}}^{2}$ 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) $a_{j}$, $b_{j}$, $j=1,2,\dots,g$, such that their intersection indices satisfy … The zeros $\lambda_{j}$, $j=1,2,\dots,2g+1$ 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 $B=\{P_{1},P_{2},\dots,P_{2g+1},P_{\infty}\}$. …
##### 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 $F(z_{0})=e^{\alpha\ln\left(1-z_{0}\right)}e^{\beta\ln\left(1+z_{0}\right)}$ 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 $x_{0} 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 $\mathbf{x}=\boldsymbol{{0}}$, 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 $\Phi$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp\left(i\Phi\right)$. …
###### §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 $\Phi$, 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. …