fractional transformations

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1: 1.12 Continued Fractions

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Fractional Transformations

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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

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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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Notes:

Reprinted in 1991.

External Links:

ISBN 0-471-01525-3, MathReview (A. E. Heins), ZentralBlatt

Referenced by:

§1.11(v), Ch.3.

Encodings:

BibTeX

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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

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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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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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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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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

BibTeX

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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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Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

Encodings:

BibTeX

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9: 15.19 Methods of Computation

… ►For $z\in ℝ$ 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 $[0,\frac{1}{2}]$. … ►When $\mathrm{\Re }z>\frac{1}{2}$ 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: 19.13 Integrals of Elliptic Integrals

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for $F(\varphi ,k)$ and $E(\varphi ,k)$, together with special cases. ►

§19.13(iii) Laplace Transforms

►For direct and inverse Laplace transforms for the complete elliptic integrals $K\left(k\right)$, $E\left(k\right)$, and $D\left(k\right)$ see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.