on finite point sets
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1: 18.2 General Orthogonal Polynomials
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Orthogonality on Finite Point Sets
►Let be a finite set of distinct points on , or a countable infinite set of distinct points on , and , , be a set of positive constants. …when is a finite set of distinct points. …2: 23.20 Mathematical Applications
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►Let denote the set of points on that are of finite order (that is, those points
for which there exists a positive integer with ), and let 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
, say, is an open set in 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.
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4: 36.12 Uniform Approximation of Integrals
§36.12 Uniform Approximation of Integrals
►§36.12(i) General Theory for Cuspoids
… ►Also, is real analytic, and for all such that all critical points coincide. … ►In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. … ►Also, and are chosen to be positive real when 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 be a point set with a limit point
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►If is a finite limit point of , then
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►Suppose is a parameter (or set of parameters) ranging over a point set (or sets) , and for each nonnegative integer
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►Similarly for finite limit point
in place of .
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►where is a finite, or infinite, limit point of .
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6: 21.7 Riemann Surfaces
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►Consider the set of points in that satisfy the equation
…This compact curve may have singular points, that is, points at which the gradient of vanishes.
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►On this surface, we choose
cycles (that is, closed oriented curves, each with at most a finite number of singular points) , , , such that their intersection indices satisfy
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►The zeros , of specify the finite branch points
, that is, points at which , on the Riemann surface.
Denote the set of all branch points by .
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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.
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►A cut domain is one from which the points on finitely many nonintersecting simple contours (§1.9(iii)) have been removed.
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►Alternatively, take to be any point in and set
where the logarithms assume their principal values.
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►(The integer may be greater than one to allow for a finite number of zero factors.)
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8: 1.4 Calculus of One Variable
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►where the supremum is over all sets of points
in the closure of , that is, with added when they are finite.
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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 , 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 -axis containing all real critical points of and is deformed outside this range so as to reach infinity along the asymptotic valleys of .
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