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fractional transformations


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1: 1.12 Continued Fractions
Fractional Transformations
2: 19.14 Reduction of General Elliptic Integrals
The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. …
3: 1.9 Calculus of a Complex Variable
Bilinear Transformation
Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …
4: Bibliography H
  • P. Henrici (1977) Applied and Computational Complex Analysis. Vol. 2: Special Functions—Integral Transforms—Asymptotics—Continued Fractions. Wiley-Interscience [John Wiley & Sons], New York.
  • 5: 15.8 Transformations of Variable
    A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. …
    6: Bibliography K
  • T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
  • 7: Bibliography J
  • W. B. Jones and W. Van Assche (1998) Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.
  • 8: Bibliography C
  • A. D. Chave (1983) Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.
  • 9: 15.19 Methods of Computation
    For z it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval [ 0 , 1 2 ] . … When z > 1 2 it is better to begin with one of the linear transformations (15.8.4), (15.8.7), or (15.8.8). …
    §15.19(v) Continued Fractions
    In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …
    10: 2.6 Distributional Methods
    §2.6(ii) Stieltjes Transform
    The Stieltjes transform of f ( t ) is defined by … Corresponding results for the generalized Stieltjes transform
    §2.6(iii) Fractional Integrals
    The Riemann–Liouville fractional integral of order μ is defined by …