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1: 1.4 Calculus of One Variable
1.4.33 𝒱 a , b ( f ) = sup j = 1 n | f ( x j ) - f ( x j - 1 ) | ,
where the supremum is over all sets of points x 0 < x 1 < < x n in the closure of ( a , b ) , that is, ( a , b ) with a , b added when they are finite. … If f ( x ) is continuous on the closure of ( a , b ) and f ( x ) is continuous on ( a , b ) , then …
2: 13.9 Zeros
Let P α denote the closure of the domain that is bounded by the parabola y 2 = 4 α ( x + α ) and contains the origin. …
3: 18.2 General Orthogonal Polynomials
It is assumed throughout this chapter that for each polynomial p n ( x ) that is orthogonal on an open interval ( a , b ) the variable x is confined to the closure of ( a , b ) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … More generally than (18.2.1)–(18.2.3), w ( x ) d x may be replaced in (18.2.1) by a positive measure d α ( x ) , where α ( x ) is a bounded nondecreasing function on the closure of ( a , b ) with an infinite number of points of increase, and such that 0 < a b x 2 n d α ( x ) < for all n . …
4: 3.7 Ordinary Differential Equations
Let ( a , b ) be a finite or infinite interval and q ( x ) be a real-valued continuous (or piecewise continuous) function on the closure of ( a , b ) . … If q ( x ) is C on the closure of ( a , b ) , then the discretized form (3.7.13) of the differential equation can be used. …
5: 1.9 Calculus of a Complex Variable
Also, the union of S and its limit points is the closure of S . …
6: 13.7 Asymptotic Expansions for Large Argument
See accompanying text
Figure 13.7.1: Regions R 1 , R 2 , R ¯ 2 , R 3 , and R ¯ 3 are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with r = | b - 2 a | . Magnify
7: 2.10 Sums and Sequences
  • (a)

    On the strip a z n , f ( z ) is analytic in its interior, f ( 2 m ) ( z ) is continuous on its closure, and f ( z ) = o ( e 2 π | z | ) as z ± , uniformly with respect to z [ a , n ] .

  • 8: 1.16 Distributions
    The closure of the set of points where ϕ 0 is called the support of ϕ . …
    9: 18.18 Sums
    In this subsection the variables x and y are not confined to the closures of the intervals of orthogonality; compare §18.2(i). …