Liouville transformation

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1: 1.13 Differential Equations

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Liouville Transformation

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Transformation to Liouville normal Form

►Equation (1.13.26) with

x\in[a,b]

may be transformed to the Liouville normal form …

2: 2.8 Differential Equations with a Parameter

… ►In Case III

f(z)

has a simple pole at

z_{0}

and

(z-z_{0})^{2}g(z)

is analytic at

z_{0}

. … ►First we apply the Liouville transformation (§1.13(iv)) to (2.8.1). …

3: Bibliography S

… ►

D. R. Smith (1986) Liouville-Green approximations via the Riccati transformation. J. Math. Anal. Appl. 116 (1), pp. 147–165.

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External Links:: ISSN 0022-247X, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §2.7(iii).
Encodings:: BibTeX

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4: 2.6 Distributional Methods

… ►

§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of

f(t)

is defined by … ►Corresponding results for the generalized Stieltjes transform … ►The Riemann–Liouville fractional integral of order

\mu

is defined by … ►where

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)

is the Mellin transform of

f

or its analytic continuation. …

5: Bibliography E

… ►

U. T. Ehrenmark (1995) The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:: ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1954b) Tables of Integral Transforms. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:: Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1385, v. 41 (1983), no. 164, pp. 779–780, v. 31 (1977), no. 138, p. 614, v. 31 (1977), no. 137, pp. 328–329, v. 26 (1972), no. 118, p. 599, v. 25 (1971), no. 113, p. 199, v. 23 (1969), no. 106, p. 468.
External Links:: MathReview (H. Kober), ZentralBlatt
Referenced by:: §1.14(viii), §10.22(v), §10.22(vi), §10.43(v), §10.43(vi), §12.12, §13.10(v), §13.10(v), §13.10(vi), §13.23(iii), §13.23(v), §15.14, §15.14, §18.17(ix), §5.13, §8.14.
Encodings:: BibTeX

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W. N. Everitt (1982) On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

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Notes:: With a loose Russian summary
External Links:: ISSN 0011-4642, MathReview (D. Hinton)
Referenced by:: §1.13(viii).
Encodings:: BibTeX

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W. N. Everitt (2005a) A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville theory, pp. 271–331.

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External Links:: Document, Link, MathReview Entry
Referenced by:: §1.18(ix), §1.18(x).
Encodings:: BibTeX

►

W. N. Everitt (2005b) Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950. In Sturm-Liouville theory, pp. 45–74.

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External Links:: Document, Link, MathReview Entry
Referenced by:: §1.18(ix), §1.18(x).
Encodings:: BibTeX

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6: Bibliography P

… ►

R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

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E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

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A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt
Referenced by:: §1.14(vii).
Encodings:: BibTeX

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A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992a) Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:: Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.
External Links:: ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).
Encodings:: BibTeX

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J. D. Pryce (1993) Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.

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External Links:: ISBN 0-19-853415-9, MathReview (L. Greenberg), ZentralBlatt
Referenced by:: §3.7(iv).
Encodings:: BibTeX

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7: Errata

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Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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8: 2.7 Differential Equations

… ►See §2.11(v) for other examples. … ►

§2.7(iii) Liouville–Green (WKBJ) Approximation

►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: ►

Liouville–Green Approximation Theorem

… ►The first of these references includes extensions to complex variables and reversions for zeros. …

9: Bibliography G

… ►

F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:: ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt
Referenced by:: §17.7(iii).
Encodings:: BibTeX

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A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:: Problem proposed by P. Smith.
External Links:: Document
Referenced by:: (9.10.14), (9.10.15), (9.10.16), §9.10(v).
Encodings:: BibTeX

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D. Gómez-Ullate, N. Kamran, and R. Milson (2010) Exceptional orthogonal polynomials and the Darboux transformation. J. Phys. A 43 (43), pp. 43016, 16 pp..

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External Links:: ISSN 1751-8113, Document, Link, MathReview (María-José Cantero)
Referenced by:: §18.36(vi).
Encodings:: BibTeX

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D. Gómez-Ullate, N. Kamran, and R. Milson (2009) An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359 (1), pp. 352–367.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Li Zhong Peng)
Referenced by:: §18.36(vi).
Encodings:: BibTeX

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W. Groenevelt (2007) Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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External Links:: ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt
Referenced by:: §18.28(xi), §18.38(iii).
Encodings:: BibTeX

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10: Bibliography T

… ►

J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

J. G. Taylor (1978) Error bounds for the Liouville-Green approximation to initial-value problems. Z. Angew. Math. Mech. 58 (12), pp. 529–537.

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External Links:: ISSN 0044-2267, Document, MathReview, ZentralBlatt
Referenced by:: §2.7(iii).
Encodings:: BibTeX

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J. G. Taylor (1982) Improved error bounds for the Liouville-Green (or WKB) approximation. J. Math. Anal. Appl. 85 (1), pp. 79–89.

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External Links:: ISSN 0022-247X, Document, MathReview (Klaus Deimling), ZentralBlatt
Referenced by:: §2.7(iii).
Encodings:: BibTeX

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N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:: ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt
Referenced by:: §2.4(i).
Encodings:: BibTeX

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F. Tu and Y. Yang (2013) Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.

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External Links:: ISSN 0002-9947, Document, FullText, MathReview (John M. Voight), ZentralBlatt
Referenced by:: §15.8(v), §15.8(v).
Encodings:: BibTeX

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