Liouville%E2%80%93Green%20approximation%20theorem
(0.003 seconds)
1—10 of 341 matching pages
1: 1.13 Differential Equations
…
►
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
…
►
§27.2(i) Definitions
… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►If , then the Euler–Fermat theorem states that … ►This is Liouville’s function. …3: Bibliography D
…
►
Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
…
►
Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
…
►
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.
…
►
Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
…
►
Uniform asymptotic expansions for Charlier polynomials.
J. Approx. Theory 112 (1), pp. 93–133.
…
4: Bibliography
…
►
Algorithm 511: CDC 6600 subroutines IBESS and JBESS for Bessel functions and , ,
.
ACM Trans. Math. Software 3 (1), pp. 93–95.
…
►
Algorithm 683: A portable FORTRAN subroutine for exponential integrals of a complex argument.
ACM Trans. Math. Software 16 (2), pp. 178–182.
►
Sturm-Liouville Theory.
Birkhäuser Verlag, Basel.
…
►
Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
…
►
Multichannel Rydberg spectroscopy of complex atoms.
Reviews of Modern Physics 68, pp. 1015–1123.
5: Bibliography S
…
►
Sturm oscillation and comparison theorems.
In Sturm-Liouville theory,
pp. 29–43.
…
►
Liouville-Green approximations via the Riccati transformation.
J. Math. Anal. Appl. 116 (1), pp. 147–165.
…
►
Liouville-Green-Olver approximations for complex difference equations.
J. Approx. Theory 96 (2), pp. 301–322.
►
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.
►
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.
…
6: 2.7 Differential Equations
…
►
§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. …7: 27.4 Euler Products and Dirichlet Series
…
►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product.
…
►
27.4.7
,
…
8: Bibliography O
…
►
An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
…
►
Error bounds for stationary phase approximations.
SIAM J. Math. Anal. 5 (1), pp. 19–29.
…
►
General connection formulae for Liouville-Green approximations in the complex plane.
Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.
►
Asymptotic approximations and error bounds.
SIAM Rev. 22 (2), pp. 188–203.
…
►
Numerical evaluation of the dilogarithm of complex argument.
Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.
…
9: 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). …10: Bibliography T
…
►
LSFBTR: A subroutine for calculating spherical Bessel transforms.
Comput. Phys. Comm. 30 (1), pp. 93–99.
…
►
Error bounds for the Liouville-Green approximation to initial-value problems.
Z. Angew. Math. Mech. 58 (12), pp. 529–537.
►
Improved error bounds for the Liouville-Green (or WKB) approximation.
J. Math. Anal. Appl. 85 (1), pp. 79–89.
…
►
Uniform asymptotic approximation of Fermi-Dirac integrals.
J. Comput. Appl. Math. 31 (3), pp. 383–387.
…
►
Numerical evaluation of exponential integral: Theis well function approximation.
Journal of Hydrology 205 (1-2), pp. 38–51.
…