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11: 18.39 Applications in the Physical Sciences
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►where is a spatial coordinate, the mass of the particle with potential energy , is the reduced Planck’s constant, and
a finite or infinite interval.
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►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being and forming a complete set.
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►This indicates that the Laguerre polynomials appearing in (18.39.29) are not classical OP’s, and in fact, even though infinite in number for fixed , do not form a complete set.
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►For the potential , corresponding to interaction of particles with like charges, there are no bound states, the continuum scattering states form a complete set for each , as discussed in Chapter 33, and their discretized versions in §18.39(iv).
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►For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge.
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12: 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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►In the complex plane has a branch point at .
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►valid when , or , .
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►Sometimes the factor is omitted.
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►For a uniform asymptotic approximation for see Temme and Olde Daalhuis (1990).
13: 3.4 Differentiation
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Two-Point Formula
… ►Three-Point Formula
… ►Four-Point Formula
… ►Five-Point Formula
… ►Six-Point Formula
…14: 18.40 Methods of Computation
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A numerical approach to the recursion coefficients and quadrature abscissas and weights
… ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let be a positive integer and define … ►Results of low ( to decimal digits) precision for are easily obtained for to . … ►The quadrature points and weights can be put to a more direct and efficient use. … ►This allows Stieltjes–Perron inversion for the , given the quadrature weights and points. …15: Bibliography S
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The zeros of special functions from a fixed point method.
SIAM J. Numer. Anal. 40 (1), pp. 114–133.
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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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On the double points of a Mathieu equation.
J. Comput. Appl. Math. 107 (1), pp. 111–125.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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16: Bibliography B
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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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A short table of the functions , from to
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Phil. Mag. Series 7 20, pp. 343–347.
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Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities.
J. Math. Mech. 17, pp. 533–559.
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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions.
SIAM J. Math. Anal. 17 (2), pp. 422–450.
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Asymptotic expansions for the coefficient functions that arise in turning-point problems.
Proc. Roy. Soc. London Ser. A 410, pp. 35–60.
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