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Liouville–Green approximation theorem

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1: 1.13 Differential Equations
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Liouville Transformation
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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
… β–ΊThis is the Sturm-Liouville form of a second order differential equation, where ′ denotes d d x . … β–ΊA regular Sturm-Liouville system will only have solutions for certain (real) values of Ξ» , these are eigenvalues. … β–Ί
Transformation to Liouville normal Form
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2: 2.7 Differential Equations
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§2.7(iii) Liouville–Green (WKBJ) Approximation
β–ΊFor irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: β–Ί
Liouville–Green Approximation Theorem
… β–ΊBy approximating … β–ΊThe first of these references includes extensions to complex variables and reversions for zeros. …
3: 2.9 Difference Equations
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§2.9(iii) Other Approximations
β–ΊFor 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). …
4: Bibliography S
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  • B. Simon (2005c) Sturm oscillation and comparison theorems. In Sturm-Liouville theory, pp. 29–43.
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    External Links:
    Document, Link, MathReview Entry
    Referenced by:
    §18.36(vi), §18.39(i).
    Encodings:
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  • 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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  • R. Spigler, M. Vianello, and F. Locatelli (1999) Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.
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    External Links:
    ISSN 0021-9045, Document, MathReview (Dumitru Acu), ZentralBlatt
    Referenced by:
    §2.9(iii).
    Encodings:
    BibTeX
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  • R. Spigler and M. Vianello (1992) 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.
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    External Links:
    ISSN 0377-0427, Document, MathReview (Wan Biao Ma), ZentralBlatt
    Referenced by:
    §2.9(iii).
    Encodings:
    BibTeX
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  • R. Spigler and M. Vianello (1997) 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. GyΕ‘ri, and G. Ladas (Eds.), pp. 567–577.
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    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    §2.9(iii).
    Encodings:
    BibTeX
    • …
    5: Bibliography G
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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
    • … β–Ί
  • M. B. Green, J. H. Schwarz, and E. Witten (1988a) Superstring Theory: Introduction, Vol. 1. 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
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    External Links:
    ISBN 0-521-35752-7, MathReview, ZentralBlatt
    Referenced by:
    §20.12(ii), 3rd item.
    Encodings:
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  • M. B. Green, J. H. Schwarz, and E. Witten (1988b) Superstring Theory: Loop Amplitudes, Anomalies and Phenomenolgy, Vol. 2. 2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
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    External Links:
    ISBN 0-521-35753-5, MathReview, ZentralBlatt
    Referenced by:
    §20.12(ii).
    Encodings:
    BibTeX
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  • C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.
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    External Links:
    Document
    Referenced by:
    item Greene et al. (1979):, Notations F, Notations F, Notations G.
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  • D. H. Greene and D. E. Knuth (1982) Mathematics for the Analysis of Algorithms. Progress in Computer Science, Vol. 1, Birkhäuser Boston, Boston, MA.
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    External Links:
    MathReview (Ernst-Erich Doberkat), ZentralBlatt
    Referenced by:
    §5.17.
    Encodings:
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    6: Bibliography O
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  • F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.
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    Notes:
    See also Olver (1997b)
    External Links:
    Document, MathReview (L. Berg), ZentralBlatt
    Referenced by:
    §2.3(iv).
    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).
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  • F. W. J. Olver (1980a) Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.
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    External Links:
    ISSN 0036-1445, Document, MathReview (L. Berg), ZentralBlatt
    Referenced by:
    §2.7(iii).
    Encodings:
    BibTeX
    • β–Ί
  • F. W. J. Olver (1980b) Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.
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    External Links:
    ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt
    Referenced by:
    §13.20(iii), §13.20(iv).
    Encodings:
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  • M. L. Overton (2001) Numerical Computing with IEEE Floating Point Arithmetic. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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    Notes:
    Including one theorem, one rule of thumb, and one hundred and one exercises
    External Links:
    ISBN 0-89871-482-6, MathReview (Jesse L. Barlow), ZentralBlatt
    Referenced by:
    §3.1(i).
    Encodings:
    BibTeX
    • 7: Bibliography T
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  • 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:
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  • N. M. Temme and A. B. Olde Daalhuis (1990) Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.
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    External Links:
    ISSN 0377-0427, Document, MathReview, ZentralBlatt
    Referenced by:
    §25.12(iii).
    Encodings:
    BibTeX
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  • A. Trellakis, A. T. Galick, and U. Ravaioli (1997) Rational Chebyshev approximation for the Fermi-Dirac integral F − 3 / 2 ⁒ ( x ) . Solid–State Electronics 41 (5), pp. 771–773.
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    External Links:
    Document
    Referenced by:
    §25.21(vii).
    Encodings:
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  • P.-H. Tseng and T.-C. Lee (1998) Numerical evaluation of exponential integral: Theis well function approximation. Journal of Hydrology 205 (1-2), pp. 38–51.
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    External Links:
    Document
    Referenced by:
    §6.18(i).
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    8: 3.8 Nonlinear Equations
    … β–ΊFor real functions f ⁒ ( x ) the sequence of approximations to a real zero ΞΎ will always converge (and converge quadratically) if either: … β–ΊInverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … β–ΊInitial approximations to the zeros can often be found from asymptotic or other approximations to f ⁒ ( z ) , or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … … β–ΊFor 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). …
    9: Bibliography D
    … β–Ί
  • S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
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    External Links:
    MathReview (M. C. Gray), ZentralBlatt
    Referenced by:
    §12.13(ii).
    Encodings:
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  • H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.
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    External Links:
    ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt
    Referenced by:
    §35.7(ii), §35.8(iv), §35.8(v).
    Encodings:
    BibTeX
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  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
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    External Links:
    ZentralBlatt
    Referenced by:
    §15.4(ii).
    Encodings:
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  • T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) 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.
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    External Links:
    ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt
    Referenced by:
    §10.41(iii), §2.8(i).
    Encodings:
    BibTeX
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  • T. M. Dunster (1994a) Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.
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    External Links:
    ISSN 1073-2772, Pdf, MathReview (J. M. H. Peters), ZentralBlatt
    Referenced by:
    §28.8(iv).
    Encodings:
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    10: Bibliography B
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  • M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.
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    External Links:
    Document
    Referenced by:
    §36.14(i).
    Encodings:
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  • L. J. Billera, C. Greene, R. Simion, and R. P. Stanley (Eds.) (1996) Formal Power Series and Algebraic Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 24, American Mathematical Society, Providence, RI.
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    External Links:
    ISBN 0-8218-0324-7, MathReview, ZentralBlatt
    Referenced by:
    §26.19.
    Encodings:
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    • … β–Ί
  • F. Bowman (1958) Introduction to Bessel Functions. Dover Publications Inc., New York.
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    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:
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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
    • … β–Ί
  • W. S. Burnside and A. W. Panton (1960) The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.
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    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
    • …
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