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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: 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.
  • 9: 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 …
    10: Bibliography B
  • W. N. Bailey (1929) Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.
  • R. Barakat and E. Parshall (1996) Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.
  • A. P. Bassom, P. A. Clarkson, and A. C. Hicks (1995) Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.
  • 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. G. C. Boyd (1990b) Stieltjes transforms and the Stokes phenomenon. Proc. Roy. Soc. London Ser. A 429, pp. 227–246.