Coulomb wave equation

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1: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

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2: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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3: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(vi) Solutions Inside the Turning Point

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4: 30.12 Generalized and Coulomb Spheroidal Functions

… ►Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation …

5: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. …

6: Bibliography F

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

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§29.11.

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Spheroidal Wave Functions. Stanford University Press, Stanford, CA.

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External Links:

MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§30.1, §30.10, 2nd item, Ch.30.

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Electromagnetic Diffraction and Propagation Problems. International Series of Monographs on Electromagnetic Waves, Vol. 1, Pergamon Press, Oxford.

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MathReview (R. F. Millar)

Referenced by:

§9.16.

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Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§9.1, Notations U, Notations V.

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Numerical treatment of Coulomb wave functions. Rev. Mod. Phys. 27 (4), pp. 399–411.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.10(ii), §33.11, §33.12(i), §33.9(ii).

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7: Bibliography H

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Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

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Document

Referenced by:

§33.7.

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Coulomb Wave Functions. In Handbuch der Physik, Bd. 41/1, S. Flügge (Ed.), pp. 408–465.

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External Links:

MathReview (G. Källén)

Referenced by:

§33.2(iii), §33.23(vii), §33.23(vii), §33.24, §33.5(ii), §33.7, Ch.33.

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Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Physics 155 (2), pp. 461–493.

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External Links:

ISSN 0003-4916, Document, MathReview

Referenced by:

§33.10(ii), §33.13, §33.22(vii).

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Bessel function expansions of Coulomb wave functions. J. Math. Phys. 26 (4), pp. 656–659.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§33.10(ii), §33.10(iii), §33.14(iv), §33.9(ii), §33.9(ii).

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The eigenvalues of the angular spheroidal wave equation. Stud. Appl. Math. 66 (3), pp. 217–240.

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ISSN 0022-2526, MathReview, ZentralBlatt

Referenced by:

§30.9(iii).

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8: Software Index

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	Open Source													With Book						Commercial
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28 Mathieu Functions and Hill’s Equation
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30 Spheroidal Wave Functions
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30.18(iii) Wave Functions														✓			✓		✓			✓			Van Buren
33 Coulomb Functions
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9: Bibliography T

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

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Notes:

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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External Links:

Document

Referenced by:

§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

Referenced by:

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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External Links:

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

Referenced by:

§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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Document, Code

Referenced by:

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

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10: 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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