oscillatory integrals

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1: 36.12 Uniform Approximation of Integrals

… ►The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. … ►

36.12.1 $$I(𝐲,k)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}kf(u;𝐲)\right)g(u,𝐲)du,$$

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Defines:

$I(𝐲,k)$: oscillatory integral (locally)

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: variable, $g(u,𝐲)$: function, $f(u,𝐲)$: function and $u(t;𝐲)$: mapping

Referenced by:

§36.13

Permalink:

http://dlmf.nist.gov/36.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

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36.12.3 $$I(𝐲,k)=\frac{\exp \left(\mathrm{i}kA(𝐲)\right)}{k^{1/(K+2)}}\sum _{m=0}^K\frac{a_m(𝐲)}{k^{m/(K+2)}}\left({\delta }_{m,0}-\left(1-{\delta }_{m,0}\right)\mathrm{i}\frac{\partial }{\partial z_m}\right){\mathrm{\Psi }}_K\left(𝐳(𝐲;k)\right)\left(1+O\left(\frac{1}{k}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\delta }_{j,k}$: Kronecker delta, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: variable, $K$: codimension, $I(𝐲,k)$: oscillatory integral, $a_m(𝐲)$: function, $A(𝐲)$: function, $𝐳(𝐲,k)$: function and $z_m(𝐲,k)$: function

Permalink:

http://dlmf.nist.gov/36.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

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36.12.11 $$I(y,k)=\frac{{\mathrm{\Delta }}^{1/4}\pi \sqrt{2}}{k^{1/3}}\exp \left(\mathrm{i}k\tilde{f}\right)\left(\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime \prime }}}+\frac{g_{-}}{\sqrt{-f_{-}^{\prime \prime }}}\right)\mathrm{Ai}\left(-k^{2/3}\mathrm{\Delta }\right)\left(1+O\left(\frac{1}{k}\right)\right)-\mathrm{i}\left(\frac{g_{+}}{\sqrt{f_{+}^{\prime \prime }}}-\frac{g_{-}}{\sqrt{-f_{-}^{\prime \prime }}}\right)\frac{{\mathrm{Ai}}^{\prime }\left(-k^{2/3}\mathrm{\Delta }\right)}{k^{1/3}{\mathrm{\Delta }}^{1/2}}\left(1+O\left(\frac{1}{k}\right)\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $k$: variable and $I(𝐲,k)$: oscillatory integral

Referenced by:

§36.13

Permalink:

http://dlmf.nist.gov/36.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(ii), §36.12 and Ch.36

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2: 3.5 Quadrature

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§3.5(vii) Oscillatory Integrals

… ► ►Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature. … … ► …

3: Bibliography V

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Newton polyhedra and estimates of oscillatory integrals. Funkcional. Anal. i Priložen. 10 (3), pp. 13–38 (Russian).

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Notes:

English translation in Functional Anal. Appl. 18 (1976), pp. 175-196.

External Links:

ISSN 0374-1990, Document, MathReview (Helmut Hamm), ZentralBlatt

Referenced by:

§36.6.

Encodings:

BibTeX

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4: Bibliography E

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A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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5: Bibliography W

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Quadrature formulas for oscillatory integral transforms. Numer. Math. 39 (3), pp. 351–360.

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External Links:

ISSN 0029-599X, Document, MathReview (A.J. Rodrigues), ZentralBlatt

Referenced by:

§10.74(vii), §3.5(vii).

Encodings:

BibTeX

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6: Bibliography D

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Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.

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External Links:

MathReview (Alan Weinstein), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

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7: Bibliography C

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A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: Application to Pearcey’s integral and its derivatives. J. Phys. A 15 (4), pp. 1179–1190.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§36.15(iii), §36.8.

Encodings:

BibTeX

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8: 36.2 Catastrophes and Canonical Integrals

§36.2 Catastrophes and Canonical Integrals

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§36.2(i) Definitions

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Canonical Integrals

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§36.2(iii) Symmetries

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9: 2.3 Integrals of a Real Variable

… ►For extensions to oscillatory integrals with more general $t$-powers and logarithmic singularities see Wong and Lin (1978) and Sidi (2010). …

10: Bibliography L

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Note on a method for computing infinite integrals of oscillatory functions. Proc. Cambridge Philos. Soc. 52 (4), pp. 764–768.

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External Links:

Document, MathReview (D. J. Hofsommer), ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

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