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 $f\in C^{\mathrm{\infty }}(I)$. … ►

Mean Value Theorem

… ►If $f(x)\in C^{n+1}[a,b]$, then …

2: 4.12 Generalized Logarithms and Exponentials

… ►For $C^{\mathrm{\infty }}$ 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 $𝚫$, and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ or analytic on $𝚫$. … ►Again, $u>0$ and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$. Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi )$, $j=1,2$, that are $C^{\mathrm{\infty }}$ on $({\alpha }_1,{\alpha }_2)$, and as $u\to \mathrm{\infty }$ … ►Also, $\psi (\xi )$ is $C^{\mathrm{\infty }}$ 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 $C^{\mathrm{\infty }}$ on $(0,{\alpha }_2)$, and as $u\to \mathrm{\infty }$ …

4: 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. …

5: 1.13 Differential Equations

… ►Let $W(z)$ satisfy (1.13.14), $\zeta (z)$ be any thrice-differentiable function of $z$, and ►

1.13.18 $$U(z)={({\zeta }^{\prime }(z))}^{1/2}W(z).$$

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Symbols:

$z$: variable, $W(\xi )$: solution, $\zeta (z)$: thrice-differentiable function and $U(z)$: solution

Permalink:

http://dlmf.nist.gov/1.13.E18

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Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

… ►

1.13.19 $$\frac{d^{2}U}{{d\zeta }^2}=\left({\dot{z}}^{2}H(z)-\frac{1}{2}\{z,\zeta \}\right)U.$$

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Symbols:

$\{z,\zeta \}$: Schwarzian derivative, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: variable, $H(z)$ , $\zeta (z)$: thrice-differentiable function, $U(z)$: solution and $\dot{z}$: derivative of $z$ with respect to $\zeta$

Permalink:

http://dlmf.nist.gov/1.13.E19

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Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

… ►

1.13.22 $$\{z,\zeta \}=-{(dz/d\zeta )}^2\{\zeta ,z\}.$$

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Symbols:

$\{z,\zeta \}$: Schwarzian derivative, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: variable and $\zeta (z)$: thrice-differentiable function

Permalink:

http://dlmf.nist.gov/1.13.E22

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Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

… ►As 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

… ►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 $C^{\mathrm{\infty }}$ 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

… ►and functions $f(x),g(x)\in C^2(a,b)$, assumed real for the moment. … ►For $𝒟(T)$ 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 $\pm \mathrm{\infty }$. … ►, $f\in C^2(X)$) of $Lf=zf$ which is moreover in $L^2\left(X\right)$. …

9: 2.3 Integrals of a Real Variable

… ►

2.3.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function

Keywords:

Laplace transform

Referenced by:

§2.3(ii)

Permalink:

http://dlmf.nist.gov/2.3.E1

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Annotations for §2.3(i), §2.3 and Ch.2

►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … ►

2.3.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}\frac{q^{(s)}(0)}{x^{s+1}},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $q(t)$: infinitely differentiable function

Keywords:

Laplace transform

Referenced by:

§2.3(i)

Permalink:

http://dlmf.nist.gov/2.3.E2

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Annotations for §2.3(i), §2.3 and Ch.2

… ►

2.3.3 $${\sigma }_n=\underset{(0,\mathrm{\infty })}{sup}(t^{-1}\ln |q^{(n)}(t)/q^{(n)}(0)|)$$

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Defines:

${\sigma }_n$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $sup$: least upper bound (supremum), $q(t)$: infinitely differentiable function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/2.3.E3

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Annotations for §2.3(i), §2.3 and Ch.2

… ►

2.3.4 $${\int }_a^b{\mathrm{e}}^{\mathrm{i}xt}q(t)dt\sim {\mathrm{e}}^{\mathrm{i}ax}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(a){\left(\frac{\mathrm{i}}{x}\right)}^{s+1}-{\mathrm{e}}^{\mathrm{i}bx}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(b){\left(\frac{\mathrm{i}}{x}\right)}^{s+1},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $q(t)$: infinitely differentiable function, $a$: left endpoint and $b$: right endpoint

Permalink:

http://dlmf.nist.gov/2.3.E4

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Annotations for §2.3(i), §2.3 and Ch.2

…

10: 3.11 Approximation Techniques

… ►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. …