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11: 36.4 Bifurcation Sets
12: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
… βΊSolutions are called roots of the equation, or zeros of . … βΊand the solutions are called fixed points of . … βΊConsider and . We have and . …13: Bibliography N
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Toda equation and its solutions in special functions.
J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole.
Ph.D. Thesis, University of Maryland, College Park, MD.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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14: 31.13 Asymptotic Approximations
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βΊFor asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999).
βΊFor asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
15: 36.5 Stokes Sets
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36.5.4
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36.5.7
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βΊFor , there are two solutions
, provided that .
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βΊThe first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9).
…For the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation
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16: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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The Askey scheme as a four-manifold with corners.
Ramanujan J. 20 (3), pp. 409–439.
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Asymptotic solution of Maxwell’s equations near caustics.
Izv. Vuz. Radiofiz. 7, pp. 1049–1056.
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17: 8 Incomplete Gamma and Related
Functions
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18: 28 Mathieu Functions and Hill’s Equation
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19: 31.18 Methods of Computation
§31.18 Methods of Computation
βΊIndependent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of ; see LaΔ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …20: 10.73 Physical Applications
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βΊand on separation of variables we obtain solutions of the form , from which a solution satisfying prescribed boundary conditions may be constructed.
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βΊSee Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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βΊThe functions , , , and arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates (§1.5(ii)):
…With the spherical harmonic defined as in §14.30(i), the solutions are of the form with , , , or , depending on the boundary conditions.
…In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential.
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