# 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 $f\in C^{n}(I)$. …When $n$ is unbounded, $f$ is infinitely differentiable on $I$ and we write $f\in C^{\infty}(I)$. …
###### Mean Value Theorem
If $f(x)\in C^{n+1}[a,b]$, then …
##### 2: 4.12 Generalized Logarithms and Exponentials
For $C^{\infty}$ generalized logarithms, see Walker (1991). …
##### 3: 2.8 Differential Equations with a Parameter
in which $\xi$ ranges over a bounded or unbounded interval or domain $\mathbf{\Delta}$, and $\psi(\xi)$ is $C^{\infty}$ or analytic on $\mathbf{\Delta}$. … Again, $u>0$ and $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$. Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that are $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and as $u\to\infty$Also, $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and $u>0$. … In the former, corresponding to any positive integer $n$ there are solutions $W_{n,j}(u,\xi)$, $j=1,2$, that are $C^{\infty}$ on $(0,\alpha_{2})$, and as $u\to\infty$
where $h=b-a$, $f\in C^{2}[a,b]$, and $a<\xi. … If in addition $f$ is periodic, $f\in C^{k}(\mathbb{R})$, 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 $C^{\infty}$ functions Gauss quadrature can be very efficient. …
##### 5: 1.13 Differential Equations
Let $W(z)$ satisfy (1.13.14), $\zeta(z)$ be any thrice-differentiable function of $z$, and
1.13.19 $\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\zeta}^{2}}=\left(\dot{z}^{2}H(z)-\tfrac{1% }{2}\left\{z,\zeta\right\}\right)U.$
1.13.21 $\left\{z,\zeta\right\}=(\ifrac{\mathrm{d}\xi}{\mathrm{d}\zeta})^{2}\left\{z,% \xi\right\}+\left\{\xi,\zeta\right\}.$
1.13.22 $\left\{z,\zeta\right\}=-(\ifrac{\mathrm{d}z}{\mathrm{d}\zeta})^{2}\left\{\zeta% ,z\right\}.$
##### 6: 1.8 Fourier Series
If a function $f(x)\in C^{2}[0,2\pi]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime}(x)$. …
##### 7: 3.7 Ordinary Differential Equations
If $q(x)$ is $C^{\infty}$ 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
converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. …
2.3.2 $\int_{0}^{\infty}e^{-xt}q(t)\,\mathrm{d}t\sim\sum_{s=0}^{\infty}\frac{q^{(s)}(% 0)}{x^{s+1}},$ $x\to+\infty$.
2.3.4 $\int_{a}^{b}e^{ixt}q(t)\,\mathrm{d}t\sim e^{iax}\sum_{s=0}^{\infty}q^{(s)}(a)% \left(\frac{i}{x}\right)^{s+1}-e^{ibx}\sum_{s=0}^{\infty}q^{(s)}(b)\left(\frac% {i}{x}\right)^{s+1},$ $x\to+\infty$.
##### 9: 3.11 Approximation Techniques
Furthermore, if $f\in C^{\infty}[-1,1]$, 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 $\alpha(x)$ is an infinitely differentiable function, then … The space $\mathcal{T}(\mathbb{R})$ of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are $O\left(|x|^{-N}\right)$ as $|x|\to\infty$ for all $N$. … Let $\mathcal{D}({\mathbb{R}}^{n})=\mathcal{D}_{n}$ be the set of all infinitely differentiable functions in $n$ variables, $\phi(x_{1},x_{2},\dots,x_{n})$, with compact support in ${\mathbb{R}}^{n}$. … For tempered distributions the space of test functions $\mathcal{T}_{n}$ is the set of all infinitely-differentiable functions $\phi$ of $n$ variables that satisfy …