fractional transformations
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1: 1.12 Continued Fractions
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Fractional Transformations
…2: 19.14 Reduction of General Elliptic Integrals
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►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations.
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3: 1.9 Calculus of a Complex Variable
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Bilinear Transformation
… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …4: Bibliography H
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Applied and Computational Complex Analysis. Vol. 2: Special Functions—Integral Transforms—Asymptotics—Continued Fractions.
Wiley-Interscience [John Wiley & Sons], New York.
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5: 15.8 Transformations of Variable
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►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.
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6: Bibliography K
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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators.
SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
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7: Bibliography J
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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.
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8: Bibliography C
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Numerical integration of related Hankel transforms by quadrature and continued fraction expansion.
Geophysics 48 (12), pp. 1671–1686.
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9: 15.19 Methods of Computation
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►For 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 .
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►When it is better to begin with one of the linear transformations (15.8.4), (15.8.7), or (15.8.8).
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§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: 19.13 Integrals of Elliptic Integrals
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►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for and , together with special cases.
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