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differentiable functions

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1: 1.4 Calculus of One Variable
If f ( n ) exists and is continuous on an interval I , then we write f C n ( I ) . …When n is unbounded, f is infinitely differentiable on I and we write f C ( I ) . …
Mean Value Theorem
If f ( x ) C n + 1 [ a , b ] , then …
2: 4.12 Generalized Logarithms and Exponentials
For C generalized logarithms, see Walker (1991). …
3: 2.8 Differential Equations with a Parameter
in which ξ ranges over a bounded or unbounded interval or domain Δ , and ψ ( ξ ) is C or analytic on Δ . … Again, u > 0 and ψ ( ξ ) is C on ( α 1 , α 2 ) . Corresponding to each positive integer n there are solutions W n , j ( u , ξ ) , j = 1 , 2 , that are C on ( α 1 , α 2 ) , and as u Also, ψ ( ξ ) is C on ( α 1 , α 2 ) , and u > 0 . … In the former, corresponding to any positive integer n there are solutions W n , j ( u , ξ ) , j = 1 , 2 , that are C on ( 0 , α 2 ) , and as u
4: 3.5 Quadrature
where h = b - a , f C 2 [ a , b ] , and a < ξ < b . … If in addition f is periodic, f C k ( ) , and the integral is taken over a period, then … Let h = 1 2 ( b - a ) and f C 4 [ a , b ] . … If f C 2 m + 2 [ a , b ] , then the remainder E n ( f ) in (3.5.2) can be expanded in the form … For C functions Gauss quadrature can be very efficient. …
5: 1.13 Differential Equations
Let W ( z ) satisfy (1.13.14), ζ ( z ) be any thrice-differentiable function of z , and
1.13.18 U ( z ) = ( ζ ( z ) ) 1 / 2 W ( z ) .
1.13.19 d 2 U d ζ 2 = ( z ˙ 2 H ( z ) - 1 2 { z , ζ } ) U .
1.13.21 { z , ζ } = ( d ξ / d ζ ) 2 { z , ξ } + { ξ , ζ } .
1.13.22 { z , ζ } = - ( d z / d ζ ) 2 { ζ , z } .
6: 1.8 Fourier Series
If a function f ( x ) C 2 [ 0 , 2 π ] is periodic, with period 2 π , then the series obtained by differentiating the Fourier series for f ( x ) term by term converges at every point to f ( x ) . …
7: 3.7 Ordinary Differential Equations
If q ( x ) is C on the closure of ( a , b ) , then the discretized form (3.7.13) of the differential equation can be used. …
8: 2.3 Integrals of a Real Variable
2.3.1 0 e - x t q ( t ) d t
converges for all sufficiently large x , and q ( t ) is infinitely differentiable in a neighborhood of the origin. …
2.3.2 0 e - x t q ( t ) d t s = 0 q ( s ) ( 0 ) x s + 1 , x + .
2.3.3 σ n = sup ( 0 , ) ( t - 1 ln | q ( n ) ( t ) / q ( n ) ( 0 ) | )
2.3.4 a b e i x t q ( t ) d t e i a x s = 0 q ( s ) ( a ) ( i x ) s + 1 - e i b x s = 0 q ( s ) ( b ) ( i x ) s + 1 , x + .
9: 3.11 Approximation Techniques
Furthermore, if f C [ - 1 , 1 ] , then the convergence of (3.11.11) is usually very rapid; compare (1.8.7) with k arbitrary. …
10: 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 …