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Liouville transformation


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1: 1.13 Differential Equations
Liouville Transformation
2: 2.8 Differential Equations with a Parameter
In Case III f ( z ) has a simple pole at z 0 and ( z z 0 ) 2 g ( z ) is analytic at z 0 . … First we apply the Liouville transformation1.13(iv)) to (2.8.1). …
3: Bibliography S
  • D. R. Smith (1986) Liouville-Green approximations via the Riccati transformation. J. Math. Anal. Appl. 116 (1), pp. 147–165.
  • 4: 2.6 Distributional Methods
    §2.6(ii) Stieltjes Transform
    The Stieltjes transform of f ( t ) is defined by … Corresponding results for the generalized Stieltjes transformThe Riemann–Liouville fractional integral of order μ is defined by … where f ( z ) is the Mellin transform of f or its analytic continuation. …
    5: Bibliography P
  • R. B. Paris (2005a) A Kummer-type transformation for a F 2 2 hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.
  • E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.
  • A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992a) Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.
  • J. D. Pryce (1993) Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.
  • 6: 2.7 Differential Equations
    See §2.11(v) for other examples. …
    §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. …
    7: Bibliography T
  • J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.
  • 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 (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.
  • F. Tu and Y. Yang (2013) Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves. Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.
  • 8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    A survey is given of the formal spectral theory of second order differential operators, typical results being presented in §1.18(i) through §1.18(viii). The various types of spectra and the corresponding eigenfunction expansions are illustrated by examples. These are based on the Liouville normal form of (1.13.29). A more precise mathematical discussion then follows in §1.18(ix). Eigenvalues and eigenfunctions of T , self-adjoint extensions of with well defined boundary conditions, and utilization of such eigenfunctions for expansion of wide classes of L 2 functions, will be the focus of the remainder of this section. For generalizations see the Weber transform (10.22.78) and an extended Bessel transform (10.22.79).Friedman (1990) provides a useful introduction to both approaches; as does the conference proceeding Amrein et al. (2005), overviewing the combination of Sturm–Liouville theory and Hilbert space theory. See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of 51 solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.
    9: Bibliography O
  • F. Oberhettinger and T. P. Higgins (1961) Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.
  • F. Oberhettinger (1990) Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.
  • F. Oberhettinger (1972) Tables of Bessel Transforms. Springer-Verlag, Berlin-New York.
  • F. Oberhettinger (1974) Tables of Mellin Transforms. Springer-Verlag, Berlin-New York.
  • 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.
  • 10: 1.15 Summability Methods
    1.15.36 h ( x , y ) = 1 2 π e y | t | e i x t F ( t ) d t ,
    where F ( t ) is the Fourier transform of f ( x ) 1.14(i)). … Moreover, lim y 0 + Φ ( x + i y ) is the Hilbert transform of f ( x ) 1.14(v)). …
    1.15.44 σ R ( θ ) = 1 2 π R R ( 1 | t | R ) e i θ t F ( t ) d t ,
    For α > 0 and x 0 , the Riemann–Liouville fractional integral of order α is defined by …