Liouville–Green approximation theorem
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1—10 of 271 matching pages
1: 1.13 Differential Equations
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Liouville Transformation
… βΊ§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
… βΊThis is the Sturm-Liouville form of a second order differential equation, where ′ denotes . … βΊA regular Sturm-Liouville system will only have solutions for certain (real) values of , these are eigenvalues. … βΊTransformation to Liouville normal Form
…2: 2.7 Differential Equations
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§2.7(iii) Liouville–Green (WKBJ) Approximation
βΊFor irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: βΊLiouville–Green Approximation Theorem
… βΊBy approximating … βΊThe 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
βΊFor 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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Sturm oscillation and comparison theorems.
In Sturm-Liouville theory,
pp. 29–43.
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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. GyΕri, and G. Ladas (Eds.),
pp. 567–577.
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5: Bibliography G
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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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Superstring Theory: Introduction, Vol. 1.
2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
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Superstring Theory: Loop Amplitudes, Anomalies and Phenomenolgy, Vol. 2.
2nd edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
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General form of the quantum-defect theory.
Phys. Rev. A 19 (4), pp. 1485–1509.
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Mathematics for the Analysis of Algorithms.
Progress in Computer Science, Vol. 1, Birkhäuser Boston, Boston, MA.
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6: Bibliography O
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Error bounds for stationary phase approximations.
SIAM J. Math. Anal. 5 (1), pp. 19–29.
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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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Asymptotic approximations and error bounds.
SIAM Rev. 22 (2), pp. 188–203.
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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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Numerical Computing with IEEE Floating Point Arithmetic.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
7: 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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Uniform asymptotic approximation of Fermi-Dirac integrals.
J. Comput. Appl. Math. 31 (3), pp. 383–387.
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Rational Chebyshev approximation for the Fermi-Dirac integral
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Solid–State Electronics 41 (5), pp. 771–773.
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Numerical evaluation of exponential integral: Theis well function approximation.
Journal of Hydrology 205 (1-2), pp. 38–51.
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8: 3.8 Nonlinear Equations
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βΊFor real functions the sequence of approximations to a real zero will always converge (and converge quadratically) if either:
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βΊInverse linear interpolation (§3.3(v)) is used to obtain the first approximation:
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βΊInitial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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βΊFor 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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9: Bibliography D
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Note on the addition theorem of parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
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Ramanujan’s master theorem for symmetric cones.
Pacific J. Math. 175 (2), pp. 447–490.
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On Vandermonde’s theorem, and some more general expansions.
Proc. Edinburgh Math. Soc. 25, pp. 114–132.
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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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Uniform asymptotic approximation of Mathieu functions.
Methods Appl. Anal. 1 (2), pp. 143–168.
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10: Bibliography B
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Waves and Thom’s theorem.
Advances in Physics 25 (1), pp. 1–26.
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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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