Liouville–Green (or WKBJ) type approximations
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1: 2.7 Differential Equations
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§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
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§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
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Error bounds for the Liouville-Green approximation to initial-value problems.
Z. Angew. Math. Mech. 58 (12), pp. 529–537.
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Improved error bounds for the Liouville-Green (or WKB) approximation.
J. Math. Anal. Appl. 85 (1), pp. 79–89.
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4: Bibliography S
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Liouville-Green approximations via the Riccati transformation.
J. Math. Anal. Appl. 116 (1), pp. 147–165.
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Liouville-Green-Olver approximations for complex difference equations.
J. Approx. Theory 96 (2), pp. 301–322.
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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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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.
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5: Bibliography O
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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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6: Bibliography D
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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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7: 3.8 Nonlinear Equations
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►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).
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8: 2.8 Differential Equations with a Parameter
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►For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978).
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9: 10.21 Zeros
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►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).
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10: Bibliography B
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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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Introduction to Bessel Functions.
Dover Publications Inc., New York.
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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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The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms.
Dover Publications, New York.
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