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1: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
…2: Bibliography M
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Hill’s Equation.
Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.
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3: Bibliography V
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A Mathieu equation for ships rolling among waves. I, II.
Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.
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Integralgleichungen für periodische Lösungen Hill’scher Differentialgleichungen.
Analysis 3 (1-4), pp. 189–203 (German).
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On the growth of convergence radii for the eigenvalues of the Mathieu equation.
Math. Nachr. 192, pp. 239–253.
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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On the rational solutions of the second Painlevé equation.
Differ. Uravn. 1 (1), pp. 79–81 (Russian).
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4: Software Index
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20 Theta Functions | |||||||||||||||||||||||||
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28 Mathieu Functions and Hill’s Equation | |||||||||||||||||||||||||
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5: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable.
2nd edition, McGraw-Hill Book Co., New York.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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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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6: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Advanced Mathematical Methods for Scientists and Engineers.
McGraw-Hill Book Co., New York.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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7: Bibliography D
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Introduction to Vector Analysis.
5th edition, Allyn and Bacon Inc., Boston, MA.
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Formes canoniques des équations confluentes de l’équation de Heun.
Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.
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Computing spectra of linear operators using the Floquet-Fourier-Hill method.
J. Comput. Phys. 219 (1), pp. 296–321.
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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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8: Bibliography W
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Linear difference equations with transition points.
Math. Comp. 74 (250), pp. 629–653.
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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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Solutions of the fifth Painlevé equation. I.
Hokkaido Math. J. 24 (2), pp. 231–267.
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Hill’s equation with a large potential.
SIAM J. Appl. Math. 45 (2), pp. 200–214.
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Asymptotic behavior of stability regions for Hill’s equation.
SIAM J. Appl. Math. 47 (5), pp. 941–958.
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9: 25.12 Polylogarithms
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►The notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):
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►The remainder of the equations in this subsection apply to principal branches.
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10: Bibliography L
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The Classical Theory of Fields.
Pergamon Press, Oxford.
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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The second Painlevé equation.
Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).
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Integrals of Bessel Functions.
McGraw-Hill Book Co., Inc., New York.
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