quadratic%20transformations
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1: 19.36 Methods of Computation
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§19.36(ii) Quadratic Transformations
►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. ►The incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6). … ►Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …2: Bibliography W
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Extension of a quadratic transformation due to Whipple with an application.
Adv. Difference Equ., pp. 2013:157, 8.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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The cubic transformation of the hypergeometric function.
Quart. J. Pure and Applied Math. 41, pp. 70–79.
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Some transformations of generalized hypergeometric series.
Proc. London Math. Soc. (2) 26 (2), pp. 257–272.
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The Airy transform.
Amer. Math. Monthly 86 (4), pp. 271–277.
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3: Bibliography G
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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Problem 72-21, Laplace transforms of Airy functions.
SIAM Rev. 15 (4), pp. 796–798.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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Fourier transforms related to a root system of rank 1.
Transform. Groups 12 (1), pp. 77–116.
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4: Bibliography M
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Rational approximations, software and test methods for sine and cosine integrals.
Numer. Algorithms 12 (3-4), pp. 259–272.
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Quadratic relations for confluent hypergeometric functions.
Tohoku Math. J. (2) 52 (4), pp. 489–513.
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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case.
J. Multivariate Anal. 77 (1), pp. 21–53.
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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case.
J. Multivariate Anal. 82 (2), pp. 299–330.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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5: Bibliography C
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Reduction theorems for elliptic integrands with the square root of two quadratic factors.
J. Comput. Appl. Math. 118 (1-2), pp. 71–85.
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Quadratic transformations of Appell functions.
SIAM J. Math. Anal. 7 (2), pp. 291–304.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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An extension of a Kummer’s quadratic transformation formula with an application.
Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.
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Optimized fast Hankel transform filters.
Geophysical Prospecting 38 (5), pp. 545–568.
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6: Bibliography D
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Integral Transforms and their Applications.
2nd edition, Applied Mathematical Sciences, Vol. 25, Springer-Verlag, New York.
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung.
Birkhäuser Verlag, Basel und Stuttgart (German).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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