Liouville%E2%80%93Green approximation theorem
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1: 1.13 Differential Equations
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Liouville Transformation
… ►§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
… ►This is the Sturm-Liouville form of a second order differential equation, where ′ denotes . … ►A regular Sturm-Liouville system will only have solutions for certain (real) values of , these are eigenvalues. … ►Transformation to Liouville normal Form
…2: 27.2 Functions
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§27.2(i) Definitions
… ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►If , then the Euler–Fermat theorem states that … ►This is Liouville’s function. …3: Bibliography D
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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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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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
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4: Bibliography
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Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions and , ,
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ACM Trans. Math. Software 3 (1), pp. 93–95.
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Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument.
ACM Trans. Math. Software 16 (2), pp. 178–182.
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Sturm-Liouville Theory.
Birkhäuser Verlag, Basel.
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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
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Multichannel Rydberg spectroscopy of complex atoms.
Reviews of Modern Physics 68, pp. 1015–1123.
5: 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
… ►By approximating … ►The first of these references includes extensions to complex variables and reversions for zeros. …6: Bibliography T
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LSFBTR: A subroutine for calculating spherical Bessel transforms.
Comput. Phys. Comm. 30 (1), pp. 93–99.
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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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Numerical evaluation of exponential integral: Theis well function approximation.
Journal of Hydrology 205 (1-2), pp. 38–51.
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7: 27.4 Euler Products and Dirichlet Series
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►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product.
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27.4.7
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8: 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). …9: 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. Győri, and G. Ladas (Eds.),
pp. 567–577.
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10: Bibliography E
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Further results on McMahon’s asymptotic approximations.
J. Phys. A 33 (36), pp. 6333–6341.
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A catalogue of Sturm-Liouville differential equations.
In Sturm-Liouville theory,
pp. 271–331.
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Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950.
In Sturm-Liouville theory,
pp. 45–74.
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