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1: 1.4 Calculus of One Variable
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Functions of Bounded Variation
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1.4.33 𝒱 a , b ⁡ ( f ) = sup j = 1 n | f ⁡ ( x j ) f ⁡ ( x j 1 ) | ,
►If 𝒱 a , b ⁡ ( f ) < , then f ⁡ ( x ) is of bounded variation on ( a , b ) . In this case, g ⁡ ( x ) = 𝒱 a , x ⁡ ( f ) and h ⁡ ( x ) = 𝒱 a , x ⁡ ( f ) f ⁡ ( x ) are nondecreasing bounded functions and f ⁡ ( x ) = g ⁡ ( x ) h ⁡ ( x ) . …
2: 1.8 Fourier Series
►Suppose that f ⁡ ( x ) is continuous and of bounded variation on [ 0 , ) . …
3: 1.14 Integral Transforms
►Suppose that f ⁡ ( t ) is absolutely integrable on ( , ) and of bounded variation in a neighborhood of t = u 1.4(v)). … ►If f ⁡ ( t ) is absolutely integrable on [ 0 , ) and of bounded variation1.4(v)) in a neighborhood of t = u , then … ►Suppose the integral (1.14.32) is absolutely convergent on the line ⁡ s = σ and f ⁡ ( x ) is of bounded variation in a neighborhood of x = u . …
4: 10.23 Sums
►provided that f ⁡ ( t ) is of bounded variation1.4(v)) on an interval [ a , b ] with 0 < a < x < b < 1 . …
5: 10.43 Integrals
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  • (b)

    g ⁡ ( x ) is piecewise continuous and of bounded variation on every compact interval in ( 0 , ) , and each of the following integrals

  • 6: 10.40 Asymptotic Expansions for Large Argument
    ►Bounds for 𝒱 z , ⁡ ( t ℓ ) are given by …
    7: 10.17 Asymptotic Expansions for Large Argument
    ►Bounds for 𝒱 z , i ⁢ ⁡ ( t ℓ ) are given by … ►The bounds (10.17.15) also apply to 𝒱 z , i ⁢ ⁡ ( t ℓ ) in the conjugate sectors. …
    8: Bibliography Y
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  • K. Yang and M. de Llano (1989) Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.
  • 9: 18.39 Applications in the Physical Sciences
    ►For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials ( α = β = 0 ) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974). …
    10: 2.3 Integrals of a Real Variable
    ►In both cases the n th error term is bounded in absolute value by x n ⁢ 𝒱 a , b ⁡ ( q ( n 1 ) ⁡ ( t ) ) , where the variational operator 𝒱 a , b is defined by …