oscillatory integrals

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11: Bibliography K

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An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives. Computer Physics Comm. 132 (1-2), pp. 142–165.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§36.15(iii).

Encodings:

BibTeX

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12: Bibliography S

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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.

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External Links:

ISSN 0036-1410, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

BibTeX

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13: 22.19 Physical Applications

… ►The subsequent time evolution is always oscillatory with period $4K\left(k\right)/\sqrt{1+2\eta }$ and modulus $k=1/\sqrt{2+{\eta }^{-1}}$: … ►There is bounded oscillatory motion near $x=0$, with period $4K\left(k\right)/\sqrt{1-\eta }$, and modulus $k=1/\sqrt{{\eta }^{-1}-1}$, for initial displacements with $|a|\le \sqrt{1/\beta }$. …

14: 11.13 Methods of Computation

… ►Other integrals that appear in §11.5(i) have highly oscillatory integrands unless $z$ is small. …

15: 2.10 Sums and Sequences

… ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). …

16: 9.16 Physical Applications

… ►The function $\mathrm{Ai}\left(x\right)$ first appears as an integral in two articles by G. … ►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. …

17: Bibliography I

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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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An integral which occurs in statistics. Proceedings of the Cambridge Philosophical Society 29, pp. 271–276.

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External Links:

Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

Encodings:

BibTeX

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Highly Oscillatory Quadrature: The Story So Far. In Numerical Mathematics and Advanced Applications, A. Bermudez de Castro and others (Eds.), pp. 97–118.

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Notes:

Proceedings of ENuMath, Santiago de Compostela (2005)

External Links:

ISBN 3-540-34287-7, MathReview, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.

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External Links:

ISSN 0002-9947, FullText, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

BibTeX

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18: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►where the integral kernel is given by … ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. … ►In what follows, integrals over the continuous parts of the spectrum will be denoted by ${𝝈}_c$, and sums over the discrete spectrum by ${𝝈}_p$, with $𝝈={𝝈}_c\cup {𝝈}_p$ denoting the full spectrum. … ►Suppose that $X$ is the whole real line in one dimension, and that $q(x)$, in (1.18.28) has (non-oscillatory) limits of $0$ at both $\pm \mathrm{\infty }$, and thus a continuous spectrum on $𝝈\ge 0$. … ►Integral transforms (10.22.78) and (10.22.79) are examples of the utility of these extensions. …

19: 18.40 Methods of Computation

… ►Given the power moments, ${\mu }_n={\int }_a^b{x}^{n}d\mu (x)$, $n=0,1,2,\mathrm{\dots }$, can these be used to find a unique $\mu (x)$, a non-decreasing, real, function of $x$, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►

18.40.6 $$\underset{\epsilon \to 0+}{lim}{\int }_a^b\frac{w(x)dx}{x^{\prime }+\mathrm{i}\epsilon -x}dx={⨍}_a^b\frac{w(x)dx}{x^{\prime }-x}-\mathrm{i}\pi w(x^{\prime }),$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${⨍}_a^b$: Cauchy principal value, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

… ►Equation (18.40.7) provides step-histogram approximations to ${\int }_a^{x}d\mu (x)$, as shown in Figure 18.40.1 for $N=12$ and $120$, shown here for the repulsive Coulomb–Pollaczek OP’s of Figure 18.39.2, with the parameters as listed therein. … ►The bottom and top of the steps at the $x_i$ are lower and upper bounds to ${\int }_a^{x_i}d\mu (x)$ as made explicit via the Chebyshev inequalities discussed by Shohat and Tamarkin (1970, pp. 42–43). … ►The PWCF $x(t,N)$ is a minimally oscillatory algebraic interpolation of the abscissas $x_{i,N},i=1,2,\mathrm{\dots },N$. …

20: 2.7 Differential Equations

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2.7.21 $$w_1(x)=f^{-1/4}(x)\exp \left(\int f^{1/2}(x)dx\right)\left(1+{ϵ}_1(x)\right),$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, ${ϵ}_j(x)$: function and $w_j(z)$: solutions

Permalink:

http://dlmf.nist.gov/2.7.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

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2.7.22 $$w_2(x)=f^{-1/4}(x)\exp \left(-\int f^{1/2}(x)dx\right)\left(1+{ϵ}_2(x)\right),$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, ${ϵ}_j(x)$: function and $w_j(z)$: solutions

Permalink:

http://dlmf.nist.gov/2.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

… ►Suppose in addition $|\int f^{1/2}(x)dx|$ is unbounded as $x\to a_1+$ and $x\to a_2-$. … ►We cannot take $f=x$ and $g=\ln x$ because $\int g{f}^{-1/2}dx$ would diverge as $x\to +\mathrm{\infty }$. … ►In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are $\frac{1}{2}\pi$ out of phase.