DLMF: Untitled Document

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D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

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J. F. Nye (2006) Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.

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External Links:: ISSN 1364-5021, Document, MathReview, ZentralBlatt
Referenced by:: §36.7(iv).
Encodings:: BibTeX

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J. F. Nye (2007) Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.

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External Links:: ISSN 1364-5021, Document, ZentralBlatt
Referenced by:: §36.7(iv).
Encodings:: BibTeX

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M. V. Berry and C. J. Howls (1990) Stokes surfaces of diffraction catastrophes with codimension three. Nonlinearity 3 (2), pp. 281–291.

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External Links:: ISSN 0951-7715, Document, MathReview (Maria Carmen Romero-Fuster), ZentralBlatt
Referenced by:: §36.2(i), §36.5(ii), §36.5(iii), §36.5(iv).
Encodings:: BibTeX

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M. V. Berry and C. J. Howls (1994) Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.

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External Links:: ISSN 0962-8444, FullText, MathReview (F. Ursell), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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M. V. Berry and C. J. Howls (2010) Axial and focal-plane diffraction catastrophe integrals. J. Phys. A 43 (37), pp. 375206, 13.

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External Links:: ISSN 1751-8113, Document, FullText, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: (36.2.28), (36.2.29), §36.2(iv).
Encodings:: BibTeX

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M. V. Berry, J. F. Nye, and F. J. Wright (1979) The elliptic umbilic diffraction catastrophe. Phil. Trans. Roy. Soc. Ser. A 291 (1382), pp. 453–484.

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External Links:: FullText
Referenced by:: §36.15(iv), §36.2(i), §36.7(ii), §36.7(iii).
Encodings:: BibTeX

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M. V. Berry and C. Upstill (1980) Catastrophe optics: Morphologies of caustics and their diffraction patterns. In Progress in Optics, E. Wolf (Ed.), Vol. 18, pp. 257–346.

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Referenced by:: §36.14(ii), Ch.36.
Encodings:: BibTeX

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… ►Diffraction catastrophes describe the (linear) wave amplitudes that smooth the geometrical caustic singularities and decorate them with interference patterns. … ►Diffraction catastrophes describe the connection between ray optics and wave optics. … ►Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. …

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M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:: Document, MathReview (Mark Adler)
Referenced by:: §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
Encodings:: BibTeX

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A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:: ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.12(iii).
Encodings:: BibTeX

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D. E. Amos (1989) Repeated integrals and derivatives of

K

Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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External Links:: ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
Referenced by:: §10.43(iii).
Encodings:: BibTeX

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V. I. Arnol’d (1986) Catastrophe Theory. 2nd edition, Springer-Verlag, Berlin.

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Notes:: Translated from the Russian by G. S. Wassermann, based on a translation by R. K. Thomas. See also Arnol’d (1992).
External Links:: ISBN 3-540-16199-6, MathReview (James Callahan), ZentralBlatt
Referenced by:: Ch.36.
Encodings:: BibTeX

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V. I. Arnol’d (1992) Catastrophe Theory. 3rd edition, Springer-Verlag, Berlin.

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Notes:: Translated from the Russian by G. S. Wassermann, Based on a translation by R. K. Thomas
External Links:: ISBN 3-540-54811-4, MathReview (Peter Giblin), ZentralBlatt
Referenced by:: V. I. Arnol’d (1986).
Encodings:: BibTeX

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FDLIBM (free C library)

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Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

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H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

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C. K. Frederickson and P. L. Marston (1992) Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

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External Links:: Document
Referenced by:: §36.14(iv).
Encodings:: BibTeX

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G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

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… ►Define a mapping

u(t;\mathbf{y})

by relating

f(u;\mathbf{y})

to the normal form (36.2.1) of

\Phi_{K}\left(t;\mathbf{x}\right)

in the following way: …with the

K+1

functions

A(\mathbf{y})

and

\mathbf{x}(\mathbf{y})

determined by correspondence of the

K+1

critical points of

f

and

\Phi_{K}

. …where

t_{j}(\mathbf{x})

,

1\leq j\leq K+1

, are the critical points of

\Phi_{K}

, that is, the solutions (real and complex) of (36.4.1). … ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes

\Psi_{K}(\mathbf{x};k)

in (36.2.10) away from

\mathbf{x}=\boldsymbol{{0}}

, in terms of canonical integrals

\Psi_{J}\left(\xi(\mathbf{x};k)\right)

for

J<K

. For example, the diffraction catastrophe

\Psi_{2}(x,y;k)

defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function

\Psi_{1}\left(\xi(x,y;k)\right)

when

k

is large, provided that

x

and

y

are not small. …

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J. Walker (1989) A drop of water becomes a gateway into the world of catastrophe optics. Scientific American 261, pp. 120–123.

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Referenced by:: §36.14(ii).
Encodings:: BibTeX

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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F. J. Wright (1980) The Stokes set of the cusp diffraction catastrophe. J. Phys. A 13 (9), pp. 2913–2928.

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External Links:: ISSN 0305-4470, Document, MathReview (I. Stewart), ZentralBlatt
Referenced by:: §36.5(ii).
Encodings:: BibTeX

…

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•

Khamis (1965) tabulates $P\left(a,x\right)$ for $a=0.05(.05)10(.1)20(.25)70$ , $0.0001\leq x\leq 250$ to 10D.

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•

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x=0(.01)2$ to 7D; also $(x+n)e^{x}E_{n}\left(x\right)$ for $n=2,3,4,10,20$ , $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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•

Pagurova (1961) tabulates $E_{n}\left(x\right)$ for $n=0(1)20$ , $x=0(.01)2(.1)10$ to 4-9S; $e^{x}E_{n}\left(x\right)$ for $n=2(1)10$ , $x=10(.1)20$ to 7D; $e^{x}E_{p}\left(x\right)$ for $p=0(.1)1$ , $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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•

Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

fold%20catastrophe

11—20 of 113 matching pages

11: Bibliography N

12: Bibliography B

13: 36.14 Other Physical Applications

14: Bibliography

15: 8 Incomplete Gamma and Related
Functions

16: 28 Mathieu Functions and Hill’s Equation

17: Bibliography F

18: 36.12 Uniform Approximation of Integrals

19: Bibliography W

20: 8.26 Tables