differentiable functions

SearchAdvancedHelp

(0.001 seconds)

1—10 of 29 matching pages

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 $\mathbf{\Delta }$, and $\psi (\xi )$ is $C^{\mathrm{\infty }}$ or analytic on $\mathbf{\Delta }$. … ►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 $$U(z)={({\zeta }^{\prime }(z))}^{1/2}W(z).$$

ⓘ

Symbols:

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

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

Symbols:

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

Permalink:

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

Encodings:

pMML, png, TeX

See also:

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

… ►

1.13.21 $$\{z,\zeta \}={(d\xi /d\zeta )}^2\{z,\xi \}+\{\xi ,\zeta \}.$$

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/1.13.E21

Encodings:

pMML, png, TeX

See also:

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

►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

…

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: 2.3 Integrals of a Real Variable

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

…

9: 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. …

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 $𝒯(ℝ)$ 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 $|x|\to \mathrm{\infty }$ for all $N$. … ►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 …