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11: 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 $f\in C^{\mathrm{\infty }}(I)$. … ►

Infinite Integrals

… ►With , the total variation of $f(x)$ on a finite or infinite interval $(a,b)$ is …where the supremum is over all sets of points in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …

12: Bibliography I

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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-0-7381-9720-3, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..

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External Links:

ISBN 978-1-5044-4602-0, Document

Referenced by:

§3.1(ii), §3.1(ii), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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13: 2.1 Definitions and Elementary Properties

… ►In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. … ►If the set $𝐗$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty }$. …Suppose $u$ is a parameter (or set of parameters) ranging over a point set (or sets) $𝐔$, and for each nonnegative integer $n$ … ►where $c$ is a finite, or infinite, limit point of $𝐗$. … ►Many properties enjoyed by Poincaré expansions (for example, multiplication) do not always carry over. …

14: 1.5 Calculus of Two or More Variables

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Infinite Integrals

►Suppose that $a,b,c$ are finite, $d$ is finite or $+\mathrm{\infty }$, and $f(x,y)$, $\partial f/\partial x$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times [c,d)$. … ►Then the double integral of $f(x,y)$ over $R$ is defined by … ►

Infinite Double Integrals

… ►Moreover, if $a,b,c,d$ are finite or infinite constants and $f(x,y)$ is piecewise continuous on the set $(a,b)\times (c,d)$, then …

15: 2.4 Contour Integrals

… ►The result in §2.3(ii) carries over to a complex parameter $z$. … ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►in which $a$ is finite, $b$ is finite or infinite, and $\omega$ is the angle of slope of $𝒫$ at $a$, that is, $lim(\mathrm{ph}\left(t-a\right))$ as $t\to a$ along $𝒫$. … ►

$z$ ranges along a ray or over an annular sector ${\theta }_1\le \theta \le {\theta }_2$, $|z|\ge Z$, where $\theta =\mathrm{ph}z$, , and $Z>0$. $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$.

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Excluding $t=a$, $\mathrm{\Re }\left({\mathrm{e}}^{\mathrm{i}\theta }p(t)-{\mathrm{e}}^{\mathrm{i}\theta }p(a)\right)$ is positive when $t\in 𝒫$, and is bounded away from zero uniformly with respect to $\theta \in [{\theta }_1,{\theta }_2]$ as $t\to b$ along $𝒫$.

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16: 1.8 Fourier Series

… ►For $f(x)$ piecewise continuous on $[a,b]$ and real $\lambda$, …(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►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 $f^{\prime }(x)$. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. … ►Suppose that $f(x)$ is continuous and of bounded variation on $[0,\mathrm{\infty })$. …

17: 2.10 Sums and Sequences

… ►Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and $f^{(2m)}(x)$ is absolutely integrable over $[a,n]$. …${\vartheta }_n$ being some number in the interval $(0,1)$. … ►

The first infinite integral in (2.10.2) converges.

… ►the last step following from $|x^t|\le 1$ when $t$ is on the interval $[-\frac{1}{2},0]$, the imaginary axis, or the small semicircle. … ►Let $\alpha$ be a constant in $(0,2\pi )$ and $P_n$ denote the Legendre polynomial of degree $n$. …

18: 2.5 Mellin Transform Methods

… ►Let $f(t)$ be a locally integrable function on $(0,\mathrm{\infty })$, that is, ${\int }_{\rho }^{T}f(t)dt$ exists for all $\rho$ and $T$ satisfying . …The domain of analyticity of $ℳf\left(z\right)$ is usually an infinite strip parallel to the imaginary axis. … ►The sum in (2.5.6) is taken over all poles of $x^{-z}ℳf\left(1-z\right)ℳh\left(z\right)$ in the strip , and it provides the asymptotic expansion of $I(x)$ for small values of $x$. … ►Let $f(t)$ and $h(t)$ be locally integrable on $(0,\mathrm{\infty })$ and …