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11: 1.16 Distributions
A test function is an infinitely differentiable function of compact support. … More generally, if α ( x ) is an infinitely differentiable function, then … The space 𝒯 ( ) of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are O ( | x | N ) as | x | for all N . … Let 𝒟 ( n ) = 𝒟 n be the set of all infinitely differentiable functions in n variables, ϕ ( x 1 , x 2 , , x n ) , with compact support in n . … For tempered distributions the space of test functions 𝒯 n is the set of all infinitely-differentiable functions ϕ of n variables that satisfy …
12: 1.6 Vectors and Vector-Valued Functions
The gradient of a differentiable scalar function f ( x , y , z ) is … The divergence of a differentiable vector-valued function 𝐅 = F 1 𝐢 + F 2 𝐣 + F 3 𝐤 is … when 𝐅 is a continuously differentiable vector-valued function. … when 𝐅 is a continuously differentiable vector-valued function. … For f and g twice-continuously differentiable functions
13: 9.11 Products
For any continuously-differentiable function f
14: 1.5 Calculus of Two or More Variables
The function f ( x , y ) is continuously differentiable if f , f / x , and f / y are continuous, and twice-continuously differentiable if also 2 f / x 2 , 2 f / y 2 , 2 f / x y , and 2 f / y x are continuous. …
1.5.6 2 f x y = 2 f y x .
If F ( x , y ) is continuously differentiable, F ( a , b ) = 0 , and F / y 0 at ( a , b ) , then in a neighborhood of ( a , b ) , that is, an open disk centered at a , b , the equation F ( x , y ) = 0 defines a continuously differentiable function y = g ( x ) such that F ( x , g ( x ) ) = 0 , b = g ( a ) , and g ( x ) = F x / F y . …
15: 1.9 Calculus of a Complex Variable
Differentiation
A function f ( z ) is complex differentiable at a point z if the following limit exists: … A function f ( z ) is said to be analytic (holomorphic) at z = z 0 if it is complex differentiable in a neighborhood of z 0 . …
16: 22.16 Related Functions
am ( x , k ) is an infinitely differentiable function of x . …
17: 2.7 Differential Equations
In a finite or infinite interval ( a 1 , a 2 ) let f ( x ) be real, positive, and twice-continuously differentiable, and g ( x ) be continuous. …
18: Bibliography W
  • P. L. Walker (1991) Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.
  • 19: 18.38 Mathematical Applications
    In consequence, expansions of functions that are infinitely differentiable on [ 1 , 1 ] in series of Chebyshev polynomials usually converge extremely rapidly. …
    20: 3.8 Nonlinear Equations
    This is an iterative method for real twice-continuously differentiable, or complex analytic, functions: …