Coulomb field

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1: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(iv) Klein–Gordon and Dirac Equations

►The relativistic motion of spinless particles in a Coulomb field, as encountered in pionic atoms and pion-nucleon scattering (Backenstoss (1970)) is described by a Klein–Gordon equation equivalent to (33.2.1); see Barnett (1981a). The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. …

2: Bibliography Y

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

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External Links:

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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3: Bibliography K

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Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).

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External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

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Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.

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External Links:

Document, MathReview (C. Gilbert), ZentralBlatt

Referenced by:

§31.17(ii).

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Sferoidalnye i kulonovskie sferoidalnye funktsii. Izdat. “Nauka”, Moscow (Russian).

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Notes:

Translated title: Spheroidal and Coulomb Spheroidal Functions

External Links:

MathReview

Referenced by:

§30.12, Ch.30, §31.13.

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Electromagnetic Fields and Relativistic Particles. International Series in Pure and Applied Physics, McGraw-Hill Book Co., New York.

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External Links:

ISBN 0-07-035264-X, MathReview (George Zipfel, Jr.)

Referenced by:

§1.17(ii), §10.73(ii).

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4: Bibliography B

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The attractive Coulomb potential polynomials. Constr. Approx. 1 (2), pp. 103–119.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (W. A. Al-Salam)

Referenced by:

§18.39(iv), §18.39(iv).

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An algorithm for regular and irregular Coulomb and Bessel functions of real order to machine accuracy. Comput. Phys. Comm. 21 (3), pp. 297–314.

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External Links:

Document

Referenced by:

§3.10(iii), §33.22(iv).

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Zero-field susceptibility of the two-dimensional Ising model near $T_c$ . Phys. Rev. Lett. 31, pp. 1409–1411.

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External Links:

Document

Referenced by:

§32.16.

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Electromagnetic Fields and Interactions. Vol. I, Blaisdell, New York.

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Notes:

Replaces Abraham & Becker, Classical Theory of Electricity and Magnetism.

Referenced by:

§19.33(iii), §19.34.

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

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Document, Code

Referenced by:

1st item, 4th item.

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5: 18.39 Applications in the Physical Sciences

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The Quantum Coulomb Problem

… ►This is Coulomb’s Law. … ►

c) Spherical Radial Coulomb Wave Functions

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The Relativistic Quantum Coulomb Problem

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The Coulomb–Pollaczek Polynomials

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6: Bibliography P

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Influence of equilibrium shear flow along the magnetic field on the resistive tearing instability. Phys. Fluids 26 (10), pp. 2966–2975.

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External Links:

ISSN 0031-9171, Document, MathReview, ZentralBlatt

Referenced by:

§11.12, §11.7(ii).

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Statistical Field Theory. Addison-Wesley, Reading, MA.

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External Links:

ISBN 0-201-05985-1, MathReview (Tony C. Dorlas), ZentralBlatt

Referenced by:

§22.19(ii).

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The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic. Philos. Mag. (7) 37, pp. 311–317.

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

§36.2(ii).

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Gauge Field Theories. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-26537-1, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§22.19(ii).

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

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External Links:

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

Referenced by:

§33.4.

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7: Errata

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )\sim {\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k}$$

Previously this formula was expressed as an equality. Since this formula expresses an asymptotic expansion, it has been corrected by using instead an $\sim$ relation.

Reported by Gergő Nemes on 2019-01-29

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The previous constraint $\mathrm{\Re }\left(b-a\right)>0$ was removed, see Fields (1966, (3)).

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33.6.5 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\left(\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)-\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}\right)$$

Originally the factor in the denominator on the right-hand side was written incorrectly as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$.

Reported by Ian Thompson on 2018-05-17

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33.11.1 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k{\left(b\right)}_k}{k!{(\pm 2\mathrm{i}\rho )}^k}$$

Originally the factor in the denominator on the right-hand side was written incorrectly as ${(\mp 2\mathrm{i}\rho )}^k$. This has been corrected to ${(\pm 2\mathrm{i}\rho )}^k$.

Reported by Ian Thompson on 2018-05-17

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A sentence was added in §8.18(ii) to refer to Nemes and Olde Daalhuis (2016). Originally §8.11(iii) was applicable for real variables $a$ and $x=\lambda a$. It has been extended to allow for complex variables $a$ and $z=\lambda a$ (and we have replaced $x$ with $z$ in the subsection heading and in Equations (8.11.6) and (8.11.7)). Also, we have added two paragraphs after (8.11.9) to replace the original paragraph that appeared there. Furthermore, the interval of validity of (8.11.6) was increased from to the sector , and the interval of validity of (8.11.7) was increased from $\lambda >1$ to the sector $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$. A paragraph with reference to Nemes (2016) has been added in §8.11(v), and the sector of validity for (8.11.12) was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$. Two new Subsections 13.6(vii), 13.18(vi), both entitled Coulomb Functions, were added to note the relationship of the Kummer and Whittaker functions to various forms of the Coulomb functions. A sentence was added in both §13.10(vi) and §13.23(v) noting that certain generalized orthogonality can be expressed in terms of Kummer functions.

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8: 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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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(iii).

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Special Functions, $q$-Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.

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External Links:

ISBN 0-8218-0524-X, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

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Statistical Field Theory: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. Vol. 2, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-37012-4; 0-521-40806-7, MathReview (C. A. Hurst), ZentralBlatt

Referenced by:

2nd item.

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Quantum Field Theory. International Series in Pure and Applied Physics, McGraw-Hill International Book Co., New York.

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Notes:

Reprinted by Dover, 2006.

External Links:

ISBN 0-07-032071-3, MathReview (V. E. Korepin)

Referenced by:

§22.19(ii).

Encodings:

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