quadratic equations
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11: Bibliography C
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Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique.
Masson et Cie, Paris (French).
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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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Elementary Differential Equations.
Clarendon Press, Oxford.
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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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12: Bibliography G
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The Computation of Special Functions by Linear Difference Equations.
In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.),
pp. 213–243.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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Linear Differential Equations and Group Theory from Riemann to Poincaré.
2nd edition, Birkhäuser Boston Inc., Boston, MA.
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Special classes of solutions of Painlevé equations.
Differ. Uravn. 18 (3), pp. 419–429 (Russian).
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Theory of Painlevé’s equations.
Differ. Uravn. 11 (11), pp. 373–376 (Russian).
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13: 10.74 Methods of Computation
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§10.74(ii) Differential Equations
►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … ►For further information, including parallel methods for solving the differential equations, see Lozier and Olver (1993). … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …14: 32.7 Bäcklund Transformations
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§32.7(ii) Second Painlevé Equation
… ►§32.7(iii) Third Painlevé Equation
… ► also has quadratic and quartic transformations. …The quadratic transformation … …15: 19.22 Quadratic Transformations
§19.22 Quadratic Transformations
… ►As , and converge quadratically to and 0, respectively, and converges to 0 faster than quadratically. … ►The equations inverse to (19.22.5) and (19.22.16) are given by …and the corresponding equations with , , and replaced by , , and , respectively. …16: 16.16 Transformations of Variables
17: 18.2 General Orthogonal Polynomials
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►It is assumed throughout this chapter that for each polynomial that is orthogonal on an open interval the variable is confined to the closure of
unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.)
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18.2.11_2
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18.2.11_6
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§18.2(vii) Quadratic Transformations
… ►Equations (18.14.3_5) and (18.14.8), both for , can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.18: 31.7 Relations to Other Functions
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►They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)).
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►equation (31.2.1) becomes Lamé’s equation with independent variable ; compare (29.2.1) and (31.2.8).
The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively.
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19: Bibliography
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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation.
Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).
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Nonlinear chains and Painlevé equations.
Phys. D 73 (4), pp. 335–351.
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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates.
Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.
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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.
J. Math. Anal. Appl. 416 (1), pp. 52–80.
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