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Liouville–Green (or WKBJ) approximation


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1: 2.7 Differential Equations
§2.7(iii) LiouvilleGreen (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:
LiouvilleGreen 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
  • J. G. Taylor (1978) Error bounds for the Liouville-Green approximation to initial-value problems. Z. Angew. Math. Mech. 58 (12), pp. 529–537.
  • J. G. Taylor (1982) Improved error bounds for the Liouville-Green (or WKB) approximation. J. Math. Anal. Appl. 85 (1), pp. 79–89.
  • 4: Bibliography S
  • D. R. Smith (1986) Liouville-Green approximations via the Riccati transformation. J. Math. Anal. Appl. 116 (1), pp. 147–165.
  • R. Spigler, M. Vianello, and F. Locatelli (1999) Liouville-Green-Olver approximations for complex difference equations. J. Approx. Theory 96 (2), pp. 301–322.
  • 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.
  • 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. Győri, and G. Ladas (Eds.), pp. 567–577.
  • 5: Bibliography O
  • 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.
  • 6: Bibliography D
  • 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.
  • 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 LiouvilleGreen 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
  • 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.
  • F. Bowman (1958) Introduction to Bessel Functions. Dover Publications Inc., New York.
  • 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.
  • 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.