# differentiable functions

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##### 1: 1.4 Calculus of One Variable
βΊIf $f^{(n)}$ 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 $f\in C^{\infty}(I)$. … βΊ
###### Mean Value Theorem
βΊIf $f(x)\in C^{n+1}[a,b]$, then …
##### 2: 4.12 Generalized Logarithms and Exponentials
βΊFor $C^{\infty}$ generalized logarithms, see Walker (1991). …
##### 3: 2.8 Differential Equations with a Parameter
βΊin which $\xi$ ranges over a bounded or unbounded interval or domain $\mathbf{\Delta}$, and $\psi(\xi)$ is $C^{\infty}$ or analytic on $\mathbf{\Delta}$. … βΊAgain, $u>0$ 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 $u\to\infty$βΊAlso, $\psi(\xi)$ is $C^{\infty}$ on $(\alpha_{1},\alpha_{2})$, and $u>0$. … βΊIn 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 $u\to\infty$
βΊwhere $h=b-a$, $f\in C^{2}[a,b]$, and $a<\xi. … βΊIf in addition $f$ is periodic, $f\in C^{k}(\mathbb{R})$, and the integral is taken over a period, then … βΊLet $h=\frac{1}{2}(b-a)$ and $f\in C^{4}[a,b]$. … βΊIf $f\in C^{2m+2}[a,b]$, then the remainder $E_{n}(f)$ in (3.5.2) can be expanded in the form … βΊFor $C^{\infty}$ functions Gauss quadrature can be very efficient. …
##### 5: 1.13 Differential Equations
βΊLet $W(z)$ satisfy (1.13.14), $\zeta(z)$ be any thrice-differentiable function of $z$, and βΊ βΊ
1.13.19 $\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\zeta}^{2}}=\left(\dot{z}^{2}H(z)-\tfrac{1% }{2}\left\{z,\zeta\right\}\right)U.$
βΊ
1.13.22 $\left\{z,\zeta\right\}=-(\ifrac{\mathrm{d}z}{\mathrm{d}\zeta})^{2}\left\{\zeta% ,z\right\}.$
βΊAs the interval $[a,b]$ is mapped, one-to-one, onto $[0,c]$ by the above definition of $t$, the integrand being positive, the inverse of this same transformation allows $\widehat{q}(t)$ to be calculated from $p,q,\rho$ in (1.13.31), $p,\rho\in C^{2}(a,b)$ and $q\in C(a,b)$. …
##### 6: 1.8 Fourier Series
βΊIf 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
βΊIf $q(x)$ is $C^{\infty}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …
##### 8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
βΊand functions $f(x),g(x)\in C^{2}(a,b)$, assumed real for the moment. … βΊFor $\mathcal{D}(T)$ we can take $C^{2}(X)$, with appropriate boundary conditions, and with compact support if $X$ is bounded, which space is dense in $L^{2}\left(X\right)$, and for $X$ unbounded require that possible non-$L^{2}$ eigenfunctions of (1.18.28), with real eigenvalues, are non-zero but bounded on open intervals, including $\pm\infty$. … βΊ, $f\in C^{2}(X)$) of $Lf=zf$ which is moreover in $L^{2}\left(X\right)$. …
##### 9: 2.3 Integrals of a Real Variable
βΊ βΊconverges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … βΊ βΊ βΊ
2.3.4 $\int_{a}^{b}e^{ixt}q(t)\,\mathrm{d}t\sim e^{iax}\sum_{s=0}^{\infty}q^{(s)}(a)% \left(\frac{i}{x}\right)^{s+1}-e^{ibx}\sum_{s=0}^{\infty}q^{(s)}(b)\left(\frac% {i}{x}\right)^{s+1},$ $x\to+\infty$.
##### 10: 3.11 Approximation Techniques
βΊFurthermore, 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. …