Coulomb wave equation

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11: Bibliography T

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Coulomb functions with complex angular momenta. Comput. Phys. Comm. 17 (4), pp. 351–355.

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Single-precision Fortran code.

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ISSN 0010-4655, Document, Code

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3rd item, 2nd item.

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Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.

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Document

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§31.17(ii).

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COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

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Notes:

Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)

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Document, Code, ZentralBlatt

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1st item, §33.23(vi), 3rd item.

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Coulomb and Bessel functions of complex arguments and order. J. Comput. Phys. 64 (2), pp. 490–509.

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ISSN 0021-9991, Document, MathReview (R. C. Varma), ZentralBlatt

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§10.74(v), §13.32(iii), §3.10(iii), §3.10(iii), §33.13, §33.23(vi), §33.26(iii), Ch.33, Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

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12: 13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). … ►

§13.28(ii) Coulomb Functions

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13: Bibliography G

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Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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ISSN 0001-0782, Document

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§33.26(ii).

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The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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ISBN 90-5699-521-9, MathReview, ZentralBlatt

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§3.6(iii), §3.6(vii).

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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ISBN 0-8176-3837-7, MathReview, ZentralBlatt

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§15.17(v).

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Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

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ISSN 0374-0641, MathReview, ZentralBlatt

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§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

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Algorithm 300: Coulomb wave functions. Comm. ACM 10 (4), pp. 244–245.

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Notes:

For remarks see same journal 12(1969), p. 692 and 16(1973), pp. 308-309 and for certification see same journal 12(1969), pp. 279-280.

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ISSN 0001-0782, Document

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§33.26(ii).

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14: Bibliography S

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The determination of phases of wave functions. Proc. Phys. Soc. 79 (6), pp. 1296–1297.

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Document, ZentralBlatt

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§33.23(vii).

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Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

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Notes:

Double-precision Fortran code.

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§33.1, §33.14(iv), 9th item.

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The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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Document

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§33.23(vii).

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

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10th item.

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

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15: Bibliography I

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The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

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§3.8(iii).

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Ordinary Differential Equations. Longmans, Green and Co., London.

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Notes:

Reprinted by Dover Publications, New York, 1944, 1956.

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MathReview, ZentralBlatt

Referenced by:

§1.13(i), §1.13(i), §1.13(i), §31.12, §31.14(i), §32.2(i), §32.2(vi), Erratum (V1.0.25) for Section 1.13.

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A First Course in the Numerical Analysis of Differential Equations. Cambridge Texts in Applied Mathematics, No. 15, Cambridge University Press, Cambridge.

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ISBN 0-521-55376-8; 0-521-55655-4, MathReview (Luis Verde-Star), ZentralBlatt

Referenced by:

§3.7(i).

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On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt

Referenced by:

§32.11(iii).

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The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, Vol. 1191, Springer-Verlag, Berlin.

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ISBN 3-540-16483-9, MathReview (Alexander A. Pankov), ZentralBlatt

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§32.11(iv), §32.4(i), Profile Alexander A. Its.

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16: Bibliography

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Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

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Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.12(i).

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Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

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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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ISSN 0033-5606, Document, MathReview (A. Erdélyi), ZentralBlatt

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§29.11.

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A new treatment of the ellipsoidal wave equation. Proc. London Math. Soc. (3) 9, pp. 21–50.

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ISSN 0024-6115, Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§29.11.

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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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MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

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17: Bibliography P

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Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.

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ZentralBlatt

Referenced by:

§32.2(iv).

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Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.

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ISSN 0962-8444, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.11(v).

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

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Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

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MathReview (L. A. Aroian), ZentralBlatt

Referenced by:

(7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).

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Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.

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Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.4.

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18: Bibliography D

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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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MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§31.12.

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Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.

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ISSN 0037-959X, MathReview, ZentralBlatt

Referenced by:

§31.12.

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The KP equation with quasiperiodic initial data. Phys. D 123 (1-4), pp. 123–152.

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Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997)

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ISSN 0167-2789, Document, MathReview (Pavel Krejčí), ZentralBlatt

Referenced by:

§21.9.

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Wave function for smooth potential and mass step. Phys. Rev. A 59 (1), pp. 107–112.

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Document

Referenced by:

§15.18.

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Coulomb wave functions with complex values of the variable and the parameters. J. Math. Phys. 40 (12), pp. 6145–6166.

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ISSN 0022-2488, Document, MathReview (Raj Wilson), ZentralBlatt

Referenced by:

§33.13.

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19: Bibliography Y

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On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

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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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Referenced by:

§18.39(iv).

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$L$-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

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Referenced by:

§18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).

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Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.

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Document, ZentralBlatt

Referenced by:

§33.10(i), §33.2(ii), §33.2(iii), §33.2(iv), §33.5(i), §33.5(iv), §33.9(ii).

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20: Bibliography B

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Coulomb wave functions for all real $\eta$ and $\rho$ . Comput. Phys. Comm. 8 (5), pp. 377–395.

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Notes:

Single-precision Fortran code for positive energies. For modification see same journal, 11(1976), p. 141.

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Referenced by:

§3.10(iii), §33.26(ii), §33.8.

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Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.

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Referenced by:

§36.14(i), §9.16.

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Waves and Thom’s theorem. Advances in Physics 25 (1), pp. 1–26.

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Referenced by:

§36.14(i).

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Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.7.

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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

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