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Liouville–Green approximation theorem

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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
By approximatingThe 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 O
  • F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.
  • 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.
  • F. W. J. Olver (1980a) Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.
  • F. W. J. Olver (1980b) 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.
  • M. L. Overton (2001) Numerical Computing with IEEE Floating Point Arithmetic. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • 4: 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.
  • N. M. Temme and A. B. Olde Daalhuis (1990) Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.
  • A. Trellakis, A. T. Galick, and U. Ravaioli (1997) Rational Chebyshev approximation for the Fermi-Dirac integral F - 3 / 2 ( x ) . Solid–State Electronics 41 (5), pp. 771–773.
  • P.-H. Tseng and T.-C. Lee (1998) Numerical evaluation of exponential integral: Theis well function approximation. Journal of Hydrology 205 (1-2), pp. 38–51.
  • 5: 3.8 Nonlinear Equations
    For real functions f ( x ) the sequence of approximations to a real zero ξ will always converge (and converge quadratically) if either: … Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: … Initial approximations to the zeros can often be found from asymptotic or other approximations to f ( z ) , or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … … 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). …
    6: Bibliography D
  • S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
  • H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.
  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
  • 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.
  • T. M. Dunster (1994a) Uniform asymptotic approximation of Mathieu functions. Methods Appl. Anal. 1 (2), pp. 143–168.
  • 7: Bibliography B
  • M. V. Berry (1976) Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.
  • 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.
  • 8: 27.2 Functions
    §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 ( a , n ) = 1 , then the Euler–Fermat theorem states that … This is Liouville’s function. …
    9: Bibliography S
  • C. W. Schelin (1983) Calculator function approximation. Amer. Math. Monthly 90 (5), pp. 317–325.
  • 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.
  • 10: 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 n = 1 λ ( n ) n - s = ζ ( 2 s ) ζ ( s ) , s > 1 ,