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1: 1.4 Calculus of One Variable

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§1.4(ii) Continuity

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Figure 1.4.1: Piecewise continuous function on $[a,b)$. Magnify

… ►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: 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 $f(x)\in C(X)\cap L^2\left(X\right)\cap 𝒟(T)$, $f(x)$ has the eigenfunction expansion, following directly from (1.18.17)–(1.18.19), … ►For $f(x)\in C(X)\cap L^2\left(X\right)\cap 𝒟(T)$, $f(x)$ has the eigenfunction expansion, analogous to that of (1.18.33), … ►More generally, for $f\in C(X)$, $x\in X$, see (1.4.24), … ►, $f\in C^2(X)$) of $Lf=zf$ which is moreover in $L^2\left(X\right)$. …

5: 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. …

6: 6.16 Mathematical Applications

… ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

7: 1.13 Differential Equations

… ► $u$ and $z$ belong to domains $U$ and $D$ respectively, the coefficients $f(u,z)$ and $g(u,z)$ are continuous functions of both variables, and for each fixed $u$ (fixed $z$) the two functions are analytic in $z$ (in $u$). … ►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 $q\in C(a,b)$. …

8: 1.17 Integral and Series Representations of the Dirac Delta

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1.17.2 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(x-a\right)\varphi (x)dx=\varphi (a),$$ $a\in ℝ$,

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $ℝ$: real line and $\varphi (x)$: continuous function

Referenced by:

§1.16(viii), §1.17(i), §1.17(ii)

Permalink:

http://dlmf.nist.gov/1.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(i), §1.17 and Ch.1

… ►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that ►

1.17.3 $$\underset{n\to \mathrm{\infty }}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\delta }_n\left(x-a\right)\varphi (x)dx=\varphi (a),$$

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Symbols:

${\delta }_n\left(x\right)$: Dirac delta sequence, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $\varphi (x)$: continuous function

Referenced by:

§1.17(ii)

Permalink:

http://dlmf.nist.gov/1.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(i), §1.17 and Ch.1

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1.17.6 $$\underset{n\to \mathrm{\infty }}{lim}\sqrt{\frac{n}{\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx=\varphi (a),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $n$: nonnegative integer and $\varphi (x)$: continuous function

Referenced by:

§1.17(i), §1.17(ii), §7.19(iii)

Permalink:

http://dlmf.nist.gov/1.17.E6

Encodings:

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See also:

Annotations for §1.17(i), §1.17 and Ch.1

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1.17.9 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}(x-a)t}dt\right)\varphi (x)dx=\varphi (a).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\varphi (x)$: continuous function

Referenced by:

§1.17(ii)

Permalink:

http://dlmf.nist.gov/1.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17 and Ch.1

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9: 3.7 Ordinary Differential Equations

… ►Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$. … ►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. …

10: 1.8 Fourier Series

… ►Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. … ►If $a_n$ and $b_n$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi ]$, then … ►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 $f^{\prime }(x)$. …