Liouville transformation

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11: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ► … ► … ► … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. …

12: Bibliography O

… ►

F. Oberhettinger and T. P. Higgins (1961) Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:: §1.14(viii).
Encodings:: BibTeX

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F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).
Encodings:: BibTeX

… ►

F. Oberhettinger (1972) Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.

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Notes:: Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.
External Links:: ISBN 0-387-05997-0, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(v), §10.43(v), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §13.10(v), §13.23(iii), §15.14, §18.17(ix), §8.14, §8.6(iii).
Encodings:: BibTeX

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F. Oberhettinger (1974) Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.

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External Links:: ISBN 0-387-06942-9, MathReview, ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §20.10(i), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii).
Encodings:: BibTeX

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F. W. J. Olver (1978) General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.

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External Links:: ISSN 0080-4614, FullText, MathReview (W. Wasow), ZentralBlatt
Referenced by:: §2.8(v), (9.13.18), §9.13(i), §9.13(i).
Encodings:: BibTeX

…

13: 18.38 Mathematical Applications

… ►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. … ►It has elegant structures, including

N

-soliton solutions, Lax pairs, and Bäcklund transformations. While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►

Radon Transform

…

14: 18.39 Applications in the Physical Sciences

… ►The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV). … ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with

L^{2}

eigenfunctions vanishing at the end points, in this case

\pm\infty

see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …

15: 1.15 Summability Methods

… ►

1.15.36

h(x,y)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}{\mathrm{e}}^{-y\left|t% \right|}{\mathrm{e}}^{-\mathrm{i}xt}F(t)\,\mathrm{d}t,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $F(x)$ : Fourier transform of $f(t)$ , $h(x,y)$ : function and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.15.E36
Encodings:: pMML, png, TeX
See also:: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

►where

F(t)

is the Fourier transform of

f(x)

(§1.14(i)). … ►Moreover,

\lim_{y\to 0+}\Im\Phi(x+\mathrm{i}y)

is the Hilbert transform of

f(x)

(§1.14(v)). … ►

1.15.44

\sigma_{R}(\theta)=\frac{1}{\sqrt{2\pi}}\int^{R}_{-R}\left(1-\frac{\left|t% \right|}{R}\right){\mathrm{e}}^{-\mathrm{i}\theta t}F(t)\,\mathrm{d}t,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $F(x)$ : Fourier transform of $f(t)$ , $\sigma_{R}(\theta)$ : function and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.15.E44
Encodings:: pMML, png, TeX
See also:: Annotations for §1.15(v), §1.15(v), §1.15 and Ch.1

… ►For

\Re\alpha>0

and

x\geq 0

, the Riemann-Liouville fractional integral of order

\alpha

is defined by …

16: Bibliography

… ►

M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-174-6, MathReview (Benno Fuchssteiner), ZentralBlatt
Referenced by:: §21.9, §21.9, §32.5, §9.16, Profile Mark J. Ablowitz.
Encodings:: BibTeX

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N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFaden
External Links:: ISBN 0-8218-4524-1, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

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W. O. Amrein, A. M. Hinz, and D. B. Pearson (Eds.) (2005) Sturm-Liouville Theory. Birkhäuser Verlag, Basel.

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Notes:: Past and present, Including papers from the International Colloquium held at the University of Geneva, Geneva, September 15–19, 2003
External Links:: ISBN 978-3-7643-7066-4; 3-7643-7066-1, Document, Link, MathReview (Anton Zettl)
Referenced by:: §1.13(viii), §1.18(x).
Encodings:: BibTeX

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W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0098-3500, Document, GAMS (module 588; package TOMS)
Referenced by:: 2nd item.
Encodings:: BibTeX

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G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

…

17: Bibliography B

… ►

W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:: ISSN 0893-9659, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

A. P. Bassom, P. A. Clarkson, and A. C. Hicks (1995) Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.

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External Links:: ISSN 0022-2526, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.10(iv), §32.2(iii), §32.7(iv), §32.8(iv).
Encodings:: BibTeX

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J. P. Boyd and A. Natarov (1998) A Sturm-Liouville eigenproblem of the fourth kind: A critical latitude with equatorial trapping. Stud. Appl. Math. 101 (4), pp. 433–455.

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External Links:: ISSN 0022-2526, Document, MathReview, ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

… ►

W. G. C. Boyd (1990b) Stieltjes transforms and the Stokes phenomenon. Proc. Roy. Soc. London Ser. A 429, pp. 227–246.

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External Links:: ISSN 0962-8444, FullText, MathReview, ZentralBlatt
Referenced by:: §2.11(v).
Encodings:: BibTeX

…

18: Bibliography L

… ►

D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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L. Lorch, M. E. Muldoon, and P. Szegő (1970) Higher monotonicity properties of certain Sturm-Liouville functions. III. Canad. J. Math. 22, pp. 1238–1265.

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External Links:: ISSN 0008-414X, MathReview (G. Sansone), ZentralBlatt
Referenced by:: §10.21(ii).
Encodings:: BibTeX

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L. Lorch, M. E. Muldoon, and P. Szegő (1972) Higher monotonicity properties of certain Sturm-Liouville functions. IV. Canad. J. Math. 24, pp. 349–368.

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External Links:: ISSN 0008-414X, MathReview (G. Sansone), ZentralBlatt
Referenced by:: §10.21(ii).
Encodings:: BibTeX

… ►

L. Lorch and P. Szegő (1963) Higher monotonicity properties of certain Sturm-Liouville functions.. Acta Math. 109, pp. 55–73.

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External Links:: ISSN 0001-5962, Document, MathReview (G. Sansone), ZentralBlatt
Referenced by:: §10.21(ii).
Encodings:: BibTeX

… ►

J. Lund (1985) Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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External Links:: ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

…

19: 1.9 Calculus of a Complex Variable

… ►

Liouville’s Theorem

… ►

Conformal Transformation

… ► ►

Bilinear Transformation

… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …