continuously differentiable
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1: 4.12 Generalized Logarithms and Exponentials
2: 18.32 OP’s with Respect to Freud Weights
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►where is real, even, nonnegative, and continuously differentiable, where increases for , and as , see Freud (1969).
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3: 1.5 Calculus of Two or More Variables
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►The function is continuously differentiable if , , and are continuous, and
twice-continuously differentiable if also , , , and are continuous.
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1.5.6
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►If is continuously differentiable, , and at , then in a neighborhood of , that is, an open disk centered at , the equation defines a continuously differentiable function such that , , and .
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►If is times continuously differentiable, then
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►Sufficient conditions for validity are: (a) and are continuous on a rectangle , ; (b) when both and are continuously differentiable and lie in .
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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: 1.6 Vectors and Vector-Valued Functions
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►when is continuously differentiable.
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►For , , and
continuously differentiable, the vectors
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►when is a continuously differentiable vector-valued function.
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►when is a continuously differentiable vector-valued function.
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►For and twice-continuously differentiable functions
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6: 18.40 Methods of Computation
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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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7: 9.11 Products
8: 30.14 Wave Equation in Oblate Spheroidal Coordinates
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►If , then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire -space.
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9: 2.7 Differential Equations
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►In a finite or infinite interval let be real, positive, and twice-continuously differentiable, and be continuous.
…has twice-continuously differentiable solutions
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