functions of matrix argument

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21: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(i) Schrödinger Equation

… ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). … ► … ►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $ϵ$ and nonnegative integer values of $\mathrm{\ell }$, but they may be continued analytically to complex arguments and order $\mathrm{\ell }$ as indicated in §33.13. … ►

Searches for resonances as poles of the $S$-matrix in the complex half-plane . See for example Csótó and Hale (1997).

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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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§13.28(iii).

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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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ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

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§18.32.

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Bessel functions $J_n(z)$ and $Y_n(z)$ of integer order and complex argument. Comput. Phys. Comm. 78 (1-2), pp. 181–189.

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Notes:

Double-precision Pascal, maximum accuracy 7D.

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ISSN 0010-4655, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

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2nd item.

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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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ISSN 0308-2105, Document, Link, MathReview (Eugenia N. Petropoulou), ZentralBlatt

Referenced by:

§14.20(ix), §14.20(ix).

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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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ISSN 0377-0427, Document, MathReview, ZentralBlatt

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§18.36(iv).

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23: Bibliography C

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Bessel functions $J_{\nu }(x)$ and $Y_{\nu }(x)$ of real order and real argument. Comput. Phys. Comm. 18 (1), pp. 133–142.

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Double-precision Fortran, maximum accuracy 16S.

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ISSN 0010-4655, Document, Code

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4th item.

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Bessel functions $I_{\nu }(x)$ and $K_{\nu }(x)$ of real order and complex argument. Comput. Phys. Comm. 24 (1), pp. 97–105.

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Double-precision Fortran, maximum accuracy 16S. For erratum see same journal 25 (1982), p. 207.

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ISSN 0010-4655, Document, Code

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2nd item.

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Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99 (1), pp. 106–114.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

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4th item, 1st item.

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$S$-matrix and $R$-matrix determination of the low-energy ${}_{}{}^5\mathrm{He}$ and ${}_{}{}^5\mathrm{Li}$ resonance parameters. Phys. Rev. C 55 (1), pp. 536–539.

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2nd item.

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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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ISBN 00-791-1158-0, MathReview (John Todd), ZentralBlatt

Referenced by:

§22.20(vii), §22.21.

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24: Bibliography F

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

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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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Contains Mathematica package for spheroidal wave functions of complex arguments with complex parameters.

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FullText

Referenced by:

§30.12, 1st item, 2nd item.

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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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MathReview (L. J. Slater), ZentralBlatt

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3rd item, 2nd item.

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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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ISSN 0036-1410, Document, FullText, MathReview (A. Reinfelds), ZentralBlatt

Referenced by:

§5.11(iii), §5.11(iii).

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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.

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MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

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25: Bibliography Y

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Tables of Airy Functions and Their Derivatives. Izdat. Nauka, Moscow (Russian).

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4th item.

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$J$-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering. J. Math. Phys. 16, pp. 410–420.

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§18.39(iv).

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Generalized Hypergeometric Functions and Laguerre Polynomials in Two Variables. In Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemporary Mathematics, Vol. 138, pp. 239–259.

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MathReview (B. M. Agrawal), ZentralBlatt

Referenced by:

§35.10.

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Japanese with English summary. Double-precision Fortran.

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ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

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Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

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MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

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26: 18.38 Mathematical Applications

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Complex Function Theory

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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 $3j$ symbol (34.2.6), with an alternative expression as a terminating ${}_3{}^{}F_2^{}$ of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. … ►The $6j$ symbol (34.4.3), with an alternative expression as a terminating balanced ${}_4{}^{}F_3^{}$ of unit argument, can be expressend in terms of Racah polynomials (18.26.3). …

27: Bibliography

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The $J$-Matrix Method. Developments and Applications. Springer-Verlag.

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ISBN 978-1-4020-6072-4

Referenced by:

§18.39(iv).

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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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ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§13.29(iii).

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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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Referenced by:

1st item, B. Döring (1966).

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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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Single and double precision, maximum accuracy 18S.

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ISSN 0098-3500, Document, Remark #1. Remark #2. GAMS (module 644; package TOMS), MathReview Entry, ZentralBlatt

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1st item, 1st item, 2nd item, 1st item.

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Spherical Bessel functions $j_n$ and $y_n$ of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.

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Single-precision Fortran, maximum accuracy 10S.

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ISSN 0010-4655, Document, Code

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1st item.

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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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ISSN 1364-5021, Document, MathReview (Wadim Zudilin), ZentralBlatt

Referenced by:

(25.12.1), (25.12.2), (25.12.3), (25.12.4), (25.12.5), (25.12.6), (25.12.7), (25.12.8), (25.12.9), §25.12(ii), §25.12(i), §25.12(i), §25.12(i), §25.12.

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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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ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

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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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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

(25.11.21), (25.11.22), (25.11.23), §25.11, §25.11(vi), §25.18(i), (25.6.15).

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The dilogarithm function of a real argument. Math. Comp. 33 (146), pp. 778–787.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item, 4th item.

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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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FullText, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.6(i), §35.6(iii).

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