differentiable functions
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11: 1.16 Distributions
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►A test function is an infinitely differentiable function of compact support.
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►More generally, if is an infinitely differentiable function, then
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►The space of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are as for all .
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►Let be the set of all infinitely differentiable functions in variables, , with compact support in .
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►For tempered distributions the space of test functions
is the set of all infinitely-differentiable functions
of variables that satisfy
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12: 1.6 Vectors and Vector-Valued Functions
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►The gradient of a differentiable scalar function
is
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►The divergence of a differentiable vector-valued function
is
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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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13: 9.11 Products
14: 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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15: 1.9 Calculus of a Complex Variable
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Differentiation
►A function is complex differentiable at a point if the following limit exists: … ►A function is said to be analytic (holomorphic) at if it is complex differentiable in a neighborhood of . …16: 22.16 Related Functions
17: 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.
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18: Bibliography W
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Infinitely differentiable generalized logarithmic and exponential functions.
Math. Comp. 57 (196), pp. 723–733.
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19: 18.38 Mathematical Applications
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►In consequence, expansions of functions that are infinitely differentiable on in series of Chebyshev polynomials usually converge extremely rapidly.
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20: 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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