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1: 31.15 Stieltjes Polynomials
The system (31.15.2) determines the z k as the points of equilibrium of n movable (interacting) particles with unit charges in a field of N particles with the charges γ j / 2 fixed at a j . … …
2: 33.22 Particle Scattering and Atomic and Molecular Spectra
§33.22 Particle Scattering and Atomic and Molecular Spectra
With e denoting here the elementary charge, the Coulomb potential between two point particles with charges Z 1 e , Z 2 e and masses m 1 , m 2 separated by a distance s is V ( s ) = Z 1 Z 2 e 2 / ( 4 π ε 0 s ) = Z 1 Z 2 α c / s , where Z j are atomic numbers, ε 0 is the electric constant, α is the fine structure constant, and is the reduced Planck’s constant. …
§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. …
3: 19.33 Triaxial Ellipsoids
If a conducting ellipsoid with semiaxes a , b , c bears an electric charge Q , then the equipotential surfaces in the exterior region are confocal ellipsoids: …and the electric capacity C = Q / V ( 0 ) is given by … Let a homogeneous magnetic ellipsoid with semiaxes a , b , c , volume V = 4 π a b c / 3 , and susceptibility χ be placed in a previously uniform magnetic field H parallel to the principal axis with semiaxis c . The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength H / ( 1 + L c χ ) , where L c is the demagnetizing factor, given in cgs units by … The same result holds for a homogeneous dielectric ellipsoid in an electric field. …
4: Bibliography I
  • IEEE (2008) IEEE Standard for Floating-Point Arithmetic. The Institute of Electrical and Electronics Engineers, Inc..
  • IEEE (2015) IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..
  • IEEE (2018) IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..
  • IEEE (2019) IEEE International Standard for Information Technology—Microprocessor Systems—Floating-Point arithmetic: IEEE Std 754-2019. The Institute of Electrical and Electronics Engineers, Inc..
  • C. Itzykson and J. Drouffe (1989) Statistical Field Theory: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. Vol. 2, Cambridge University Press, Cambridge.
  • 5: 14.31 Other Applications
    §14.31(i) Toroidal Functions
    Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). …
    §14.31(ii) Conical Functions
    These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …
    6: 15.18 Physical Applications
    More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).
    7: Bibliography K
  • E. Kanzieper (2002) Replica field theories, Painlevé transcendents, and exact correlation functions. Phys. Rev. Lett. 89 (25), pp. (250201–1)–(250201–4).
  • A. V. Kashevarov (1998) The second Painlevé equation in electric probe theory. Some numerical solutions. Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).
  • A. V. Kashevarov (2004) The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.
  • R. P. Kerr (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.
  • E. J. Konopinski (1981) Electromagnetic Fields and Relativistic Particles. International Series in Pure and Applied Physics, McGraw-Hill Book Co., New York.
  • 8: 22.19 Physical Applications
    With appropriate scalings, Newton’s equation of motion for a pendulum with a mass in a gravitational field constrained to move in a vertical plane at a fixed distance from a fulcrum is … plays a prototypal role in classical mechanics (Lawden (1989, §5.2)), quantum mechanics (Schulman (1981, Chapter 29)), and quantum field theory (Pokorski (1987, p. 203), Parisi (1988, §14.6)). … We consider the case of a particle of mass 1, initially held at rest at displacement a from the origin and then released at time t = 0 . … Such solutions include standing or stationary waves, periodic cnoidal waves, and single and multi-solitons occurring in diverse physical situations such as water waves, optical pulses, quantum fluids, and electrical impulses (Hasegawa (1989), Carr et al. (2000), Kivshar and Luther-Davies (1998), and Boyd (1998, Appendix D2.2)). …
    9: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near T c . Phys. Rev. Lett. 31, pp. 1409–1411.
  • R. Becker and F. Sauter (1964) Electromagnetic Fields and Interactions. Vol. I, Blaisdell, New York.
  • M. V. Berry (1975) Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces. J. Phys. A 8 (4), pp. 566–584.
  • I. Bloch, M. H. Hull, A. A. Broyles, W. G. Bouricius, B. E. Freeman, and G. Breit (1951) Coulomb functions for reactions of protons and alpha-particles with the lighter nuclei. Rev. Modern Physics 23 (2), pp. 147–182.
  • 10: 9.16 Physical Applications
    An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of Ai ( x ) . …