Liouville–Green (or WKBJ) approximation

SearchAdvancedHelp

(0.002 seconds)

10 matching pages

1: 2.7 Differential Equations

… ►

§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. …

2: 2.9 Difference Equations

… ►

§2.9(iii) Other Approximations

►For asymptotic approximations to solutions of second-order difference equations analogous to the Liouville–Green (WKBJ) approximation for differential equations (§2.7(iii)) see Spigler and Vianello (1992, 1997) and Spigler et al. (1999). …

3: Bibliography T

… ►

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

ⓘ

External Links:

ISSN 0044-2267, Document, MathReview, ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

►

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

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (Klaus Deimling), ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

…

4: Bibliography S

… ►

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

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.7(iii).

Encodings:

BibTeX

… ►

Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.

ⓘ

External Links:

ISSN 0021-9045, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

►

Liouville-Green approximations for a class of linear oscillatory difference equations of the second order. J. Comput. Appl. Math. 41 (1-2), pp. 105–116.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Wan Biao Ma), ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

►

A Survey on the Liouville-Green (WKB) Approximation for Linear Difference Equations of the Second Order. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 567–577.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§2.9(iii).

Encodings:

BibTeX

…

5: Bibliography O

… ►

General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.

ⓘ

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

…

6: Bibliography D

… ►

Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

ⓘ

External Links:

ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

Encodings:

BibTeX

…

7: 3.8 Nonlinear Equations

… ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). …

8: 2.8 Differential Equations with a Parameter

… ►For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978). …

9: 10.21 Zeros

… ►For describing the distribution of complex zeros by methods based on the Liouville–Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013). …

10: Bibliography B

… ►

Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.

ⓘ

External Links:

ISBN 0-8218-0324-7, MathReview, ZentralBlatt

Referenced by:

§26.19.

Encodings:

BibTeX

… ►

Introduction to Bessel Functions. Dover Publications Inc., New York.

ⓘ

Notes:

Unaltered republication of the 1938 edition, published by Longmans, Green and Co., London-New York.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.73(iii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

… ►

The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.

ⓘ

Notes:

Two volumes. A reproduction of the seventh edition [vol. 1, Longmans, Green, London 1912; vol. 2, Longmans, Green, London 1928].

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iv).

Encodings:

BibTeX

…