polynomial%20solutions
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1: 3.8 Nonlinear Equations
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§3.8(iv) Zeros of Polynomials
►The polynomial … ►Example. Wilkinson’s Polynomial
… ►Consider and . We have and . …2: 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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Rational solutions of Painlevé equations.
Stud. Appl. Math. 61 (1), pp. 31–53.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Perturbation solutions of the ellipsoidal wave equation.
Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.
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3: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Lamé polynomial solutions to some elliptic crack and punch problems.
Internat. J. Engrg. Sci. 16 (8), pp. 551–563.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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The linear differential equation whose solutions are the products of solutions of two given differential equations.
J. Math. Anal. Appl. 98 (1), pp. 130–147.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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4: 18.39 Applications in the Physical Sciences
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►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form
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►The same solutions as in paragraph c), above, appear frequently in the literature in terms of associated Laguerre polynomials, which are referred to here as associated Coulomb–Laguerre polynomials to avoid confusion with the more recent meaning of ‘associated’ of §18.30.
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►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry.
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►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with protons, involve Laguerre polynomials of fractional index.
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The Coulomb–Pollaczek Polynomials
…5: Bibliography V
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Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
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Integral representations for products of Lamé functions by use of fundamental solutions.
SIAM J. Math. Anal. 15 (3), pp. 559–569.
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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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Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials.
J. Comput. Appl. Math. 213 (2), pp. 488–500.
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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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6: 32.8 Rational Solutions
§32.8 Rational Solutions
… ►Special rational solutions of are … ►These solutions have the form … ►These rational solutions have the form … ►7: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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On a unified approach to transformations and elementary solutions of Painlevé equations.
J. Math. Phys. 23 (11), pp. 2033–2042.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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8: 36.5 Stokes Sets
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36.5.4
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36.5.7
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►For , there are two solutions
, provided that .
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►The 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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9: Bibliography W
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Global asymptotics of the Meixner polynomials.
Asymptotic Analysis 75 (3-4), pp. 211–231.
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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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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials.
Ph.D. Thesis, University of Wisconsin, Madison, WI.
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Asymptotic solutions of a fourth order differential equation.
Stud. Appl. Math. 118 (2), pp. 133–152.
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10: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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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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