fractional transformations
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11: 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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§19.13(iii) Laplace Transforms
►For direct and inverse Laplace transforms for the complete elliptic integrals , , and see Prudnikov et al. (1992a, §3.31) and Prudnikov et al. (1992b, §§3.29 and 4.3.33), respectively.12: Errata
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►The specific updates to Chapter 18 include some results for general orthogonal polynomials including quadratic transformations, uniqueness of orthogonality measure and completeness, moments, continued fractions, and some special classes of orthogonal polynomials.
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13: Bibliography M
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Exact remainders for asymptotic expansions of fractional integrals.
J. Inst. Math. Appl. 24 (2), pp. 139–147.
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An Introduction to the Fractional Calculus and Fractional Differential Equations.
A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.
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Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions.
Ramanujan J. 6 (1), pp. 7–149.
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A new Stirling series as continued fraction.
Numer. Algorithms 56 (1), pp. 17–26.
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A continued fraction approximation of the gamma function.
J. Math. Anal. Appl. 402 (2), pp. 405–410.
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14: Bibliography S
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The Laplace Transform: Theory and Applications.
Undergraduate Texts in Mathematics, Springer-Verlag, New York.
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Numerical evaluation of the Hankel transform.
Comput. Phys. Comm. 116 (2-3), pp. 278–294.
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A code to evaluate modified Bessel functions based on the continued fraction method.
Comput. Phys. Comm. 105 (2-3), pp. 263–272.
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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms.
J. Comput. Phys. 100 (2), pp. 294–296.
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Root-rational-fraction package for exact calculation of vector-coupling coefficients.
Comput. Phys. Comm. 21 (2), pp. 195–205.
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15: 18.17 Integrals
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§18.17(iv) Fractional Integrals
►Jacobi
… ►Laguerre
… ►§18.17(v) Fourier Transforms
… ►§18.17(vi) Laplace Transforms
…16: Bibliography L
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Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions.
Comput. Phys. Comm. 99 (2-3), pp. 297–306.
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Generating Bessel functions in Mie scattering calculations using continued fractions.
Applied Optics 15 (3), pp. 668–671.
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Continued Fractions with Applications.
Studies in Computational Mathematics, North-Holland Publishing Co., Amsterdam.
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Two index laws for fractional integrals and derivatives.
J. Austral. Math. Soc. 14, pp. 385–410.
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Bessel transforms and rational extrapolation.
Numer. Math. 47 (1), pp. 1–14.
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17: 15.9 Relations to Other Functions
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►The Jacobi transform is defined as
…with inverse
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►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions.
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►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers.
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18: Bibliography T
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LSFBTR: A subroutine for calculating spherical Bessel transforms.
Comput. Phys. Comm. 30 (1), pp. 93–99.
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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions.
Quart. Appl. Math. 43 (1), pp. 103–123.
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COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments.
Comput. Phys. Comm. 36 (4), pp. 363–372.
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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”.
Comput. Phys. Comm. 159 (3), pp. 241–242.
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Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves.
Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.
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19: 16.4 Argument Unity
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§16.4(iii) Identities
… ►The basic transformation is given by … ►A different type of transformation is that of Whipple: … ►§16.4(iv) Continued Fractions
►For continued fractions for ratios of functions with argument unity, see Cuyt et al. (2008, pp. 315–317). …20: 10.43 Integrals
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