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21: 33.22 Particle Scattering and Atomic and Molecular Spectra
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►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, and , or and , to determine the scattering -matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951).
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►The Coulomb functions given in this chapter are most commonly evaluated for real values of , , , and nonnegative integer values of , but they may be continued analytically to complex arguments and order as indicated in §33.13.
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Searches for resonances as poles of the -matrix in the complex half-plane . See for example Csótó and Hale (1997).
22: Bibliography D
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Detection of the density matrix through optical homodyne tomography without filtered back projection.
Phys. Rev. A 50 (5), pp. 4298–4302.
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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory.
Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
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Bessel functions and of integer order and complex argument.
Comput. Phys. Comm. 78 (1-2), pp. 181–189.
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Conical functions of purely imaginary order and argument.
Proc. Roy. Soc. Edinburgh Sect. A 143 (5), pp. 929–955.
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A survey on orthogonal matrix polynomials satisfying second order differential equations.
J. Comput. Appl. Math. 178 (1-2), pp. 169–190.
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23: Bibliography C
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Bessel functions and of real order and real argument.
Comput. Phys. Comm. 18 (1), pp. 133–142.
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Bessel functions and of real order and complex argument.
Comput. Phys. Comm. 24 (1), pp. 97–105.
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Numerical evaluation of Airy functions with complex arguments.
J. Comput. Phys. 99 (1), pp. 106–114.
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-matrix and -matrix determination of the low-energy and resonance parameters.
Phys. Rev. C 55 (1), pp. 536–539.
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Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period.
National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.
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24: Bibliography F
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments.
National Physical Laboratory Mathematical Tables, Vol. 4.
Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
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Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument.
SIAM J. Math. Anal. 23 (2), pp. 505–511.
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Fast computation of incomplete elliptic integral of first kind by half argument transformation.
Numer. Math. 116 (4), pp. 687–719.
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Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond.
In Recent Perspectives in Random Matrix Theory and Number Theory,
London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.
25: 18.38 Mathematical Applications
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Random Matrix Theory
►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. … ►The symbol (34.2.6), with an alternative expression as a terminating of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►The symbol (34.4.3), with an alternative expression as a terminating balanced of unit argument, can be expressend in terms of Racah polynomials (18.26.3). …26: Bibliography
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Tables of for Complex Argument.
Pergamon Press, New York.
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The -Matrix Method. Developments and Applications.
Springer-Verlag.
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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule.
Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.
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A subroutine package for Bessel functions of a complex argument and nonnegative order.
Technical Report
Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.
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Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order.
ACM Trans. Math. Software 12 (3), pp. 265–273.
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27: Bibliography Y
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-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering.
J. Math. Phys. 16, pp. 410–420.
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Computation of Kummer functions for large argument
by using the -method.
Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).
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28: Bibliography M
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The dilogarithm function for complex argument.
Proc. Roy. Soc. London Ser. A 459, pp. 2807–2819.
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Calculation of the modified Bessel functions of the second kind with complex argument.
Math. Comp. 20 (95), pp. 407–412.
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Derivatives of the Hurwitz zeta function for rational arguments.
J. Comput. Appl. Math. 100 (2), pp. 201–206.
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The dilogarithm function of a real argument.
Math. Comp. 33 (146), pp. 778–787.
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Latent roots and matrix variates: A review of some asymptotic results.
Ann. Statist. 6 (1), pp. 5–33.
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