continuously%20differentiable
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1: 12.6 Continued Fraction
§12.6 Continued Fraction
►For a continued-fraction expansion of the ratio see Cuyt et al. (2008, pp. 340–341).2: 1.12 Continued Fractions
3: 10.55 Continued Fractions
§10.55 Continued Fractions
►For continued fractions for and see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).4: 1.4 Calculus of One Variable
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►If exists and is continuous on an interval , then we write .
When , is continuously
differentiable on .
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►In particular, absolute continuity occurs if the function is differentiable, with continuous.
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►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point.
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5: 18.40 Methods of Computation
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►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis.
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►In what follows we consider only the simple, illustrative, case that is continuously differentiable so that , with real, positive, and continuous on a real interval The strategy will be to: 1) use the moments to determine the recursion coefficients of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas and weights (or Christoffel numbers) from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32).
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►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to and evaluating these on the real axis in regions of higher pole density that those of the approximating function.
Results of low ( to decimal digits) precision for are easily obtained for to .
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►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows:
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6: Foreword
7: 3.8 Nonlinear Equations
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►This is an iterative method for real twice-continuously differentiable, or complex analytic, functions:
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►We continue with and either or , depending which of and is of opposite sign to , and so on.
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3.8.15
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►Consider and .
We have and .
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