differentiable

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11: 3.5 Quadrature

… ►where $h=b-a$, $f\in C^2[a,b]$, and . … ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then … ►Let $h=\frac{1}{2}(b-a)$ and $f\in C^4[a,b]$. … ►If $f\in C^{2m+2}[a,b]$, then the remainder $E_n(f)$ in (3.5.2) can be expanded in the form … ►For $C^{\mathrm{\infty }}$ functions Gauss quadrature can be very efficient. …

12: 1.16 Distributions

… ►A test function is an infinitely differentiable function of compact support. … ►More generally, if $\alpha (x)$ is an infinitely differentiable function, then … ►Suppose $f(x)$ is infinitely differentiable except at $x_0$, where left and right derivatives of all orders exist, and … ►Let $𝒟({ℝ}^n)={𝒟}_n$ be the set of all infinitely differentiable functions in $n$ variables, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$, with compact support in ${ℝ}^n$. … ►For tempered distributions the space of test functions ${𝒯}_n$ is the set of all infinitely-differentiable functions $\varphi$ of $n$ variables that satisfy …

13: 2.4 Contour Integrals

… ►On the interval let $q(t)$ be differentiable and ${\mathrm{e}}^{-ct}q(t)$ be absolutely integrable, where $c$ is a real constant. … ►Assume also (2.4.4) is differentiable. …

14: 10.20 Uniform Asymptotic Expansions for Large Order

… ►that is infinitely differentiable on the interval , including $z=1$. … ►Each of the coefficients $A_k(\zeta )$, $B_k(\zeta )$, $C_k(\zeta )$, and $D_k(\zeta )$, $k=0,1,2,\mathrm{\dots }$, is real and infinitely differentiable on the interval . …

15: Bibliography W

… ►

Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

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§4.12.

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16: 9.11 Products

… ►For any continuously-differentiable function $f$ …

17: 18.40 Methods of Computation

… ►In what follows we consider only the simple, illustrative, case that $\mu (x)$ is continuously differentiable so that $d\mu (x)=w(x)dx$, with $w(x)$ real, positive, and continuous on a real interval $[a,b].$ The strategy will be to: 1) use the moments to determine the recursion coefficients ${\alpha }_n,{\beta }_n$ of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas $x_i$ and weights (or Christoffel numbers) $w_i$ from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). …

18: 30.14 Wave Equation in Oblate Spheroidal Coordinates

… ►If $b_1=b_2=0$, then the function (30.13.8) is a twice-continuously differentiable solution of (30.13.7) in the entire $(x,y,z)$-space. …

19: 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. …has twice-continuously differentiable solutions …

20: 3.11 Approximation Techniques

… ►If $f$ is continuously differentiable on $[-1,1]$, then with … ►Furthermore, if $f\in C^{\mathrm{\infty }}[-1,1]$, then the convergence of (3.11.11) is usually very rapid; compare (1.8.7) with $k$ arbitrary. …