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1: 10.22 Integrals
§10.22(ii) Integrals over Finite Intervals
2: 10.43 Integrals
§10.43(ii) Integrals over the Intervals ( 0 , x ) and ( x , )
3: 2.8 Differential Equations with a Parameter
and for simplicity ξ is assumed to range over a finite or infinite interval ( α 1 , α 2 ) with α 1 < 0 , α 2 > 0 . …
4: 1.4 Calculus of One Variable
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. …
5: 2.1 Definitions and Elementary Properties
If c is a finite limit point of X , then … For (2.1.14) X can be the positive real axis or any unbounded sector in of finite angle. … Similarly for finite limit point c in place of . … where c is a finite, or infinite, limit point of X . … Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …
6: 24.17 Mathematical Applications
Let 0 h 1 and a , m , and n be integers such that n > a , m > 0 , and f ( m ) ( x ) is absolutely integrable over [ a , n ] . …
Calculus of Finite Differences
Let 𝒮 n denote the class of functions that have n - 1 continuous derivatives on and are polynomials of degree at most n in each interval ( k , k + 1 ) , k . …
7: 2.4 Contour Integrals
The result in §2.3(ii) carries over to a complex parameter z . … is seen to converge absolutely at each limit, and be independent of σ [ c , ) . … in which a is finite, b is finite or infinite, and ω is the angle of slope of 𝒫 at a , that is, lim ( ph ( t - a ) ) as t a along 𝒫 . …
  • (b)

    z ranges along a ray or over an annular sector θ 1 θ θ 2 , | z | Z , where θ = ph z , θ 2 - θ 1 < π , and Z > 0 . I ( z ) converges at b absolutely and uniformly with respect to z .

  • (c)

    Excluding t = a , ( e i θ p ( t ) - e i θ p ( a ) ) is positive when t 𝒫 , and is bounded away from zero uniformly with respect to θ [ θ 1 , θ 2 ] as t b along 𝒫 .

  • 8: 1.5 Calculus of Two or More Variables
    Finite Integrals
    Suppose that a , b , c are finite, d is finite or + , and f ( x , y ) , f / x are continuous on the partly-closed rectangle or infinite strip [ a , b ] × [ c , d ) . … Then the double integral of f ( x , y ) over R is defined by … Moreover, if a , b , c , d are finite or infinite constants and f ( x , y ) is piecewise continuous on the set ( a , b ) × ( c , d ) , then … Finite and infinite integrals can be defined in a similar way. …
    9: 1.8 Fourier Series
    For f ( x ) piecewise continuous on [ a , b ] and real λ , …(1.8.10) continues to apply if either a or b or both are infinite and/or f ( x ) has finitely many singularities in ( a , b ) , provided that the integral converges uniformly (§1.5(iv)) at a , b , and the singularities for all sufficiently large λ . … Let f ( x ) be an absolutely integrable function of period 2 π , and continuous except at a finite number of points in any bounded interval. … Suppose that f ( x ) is twice continuously differentiable and f ( x ) and | f ′′ ( x ) | are integrable over ( - , ) . … Suppose that f ( x ) is continuous and of bounded variation on [ 0 , ) . …
    10: 2.3 Integrals of a Real Variable
    If, in addition, q ( t ) is infinitely differentiable on [ 0 , ) and … assume a and b are finite, and q ( t ) is infinitely differentiable on [ a , b ] . … Since q ( t ) need not be continuous (as long as the integral converges), the case of a finite integration range is included. … oscillate rapidly and cancel themselves over most of the range. … If p ( b ) is finite, then both endpoints contribute: …