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1: 14.31 Other Applications
§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)). …
2: 19.18 Derivatives and Differential Equations
The next four differential equations apply to the complete case of R F and R G in the form R - a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). … Similarly, the function u = R - a ( 1 2 , 1 2 ; x + i y , x - i y ) satisfies an equation of axially symmetric potential theory: …
3: 19.33 Triaxial Ellipsoids
§19.33(ii) Potential of a Charged Conducting Ellipsoid
4: Bibliography S
  • I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.
  • C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..
  • 5: 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)).
    6: Bibliography B
  • M. N. Barber and B. W. Ninham (1970) Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.
  • M. V. Berry (1966) Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.
  • M. V. Berry (1969) Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.
  • A. Bhattacharjie and E. C. G. Sudarshan (1962) A class of solvable potentials. Nuovo Cimento (10) 25, pp. 864–879.
  • J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow (2001) Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.
  • 7: Bibliography D
  • H. Davenport (2000) Multiplicative Number Theory. 3rd edition, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York.
  • N. G. de Bruijn (1981) Pólya’s Theory of Counting. In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.), pp. 144–184.
  • L. Dekar, L. Chetouani, and T. F. Hammann (1999) Wave function for smooth potential and mass step. Phys. Rev. A 59 (1), pp. 107–112.
  • J. B. Dence and T. P. Dence (1999) Elements of the Theory of Numbers. Harcourt/Academic Press, San Diego, CA.
  • L. E. Dickson (1919) History of the Theory of Numbers (3 volumes). Carnegie Institution of Washington, Washington, D.C..
  • 8: 23.21 Physical Applications
    §23.21 Physical Applications
    In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form ( 1 - x 2 ) ( 1 - k 2 x 2 ) . The Weierstrass function plays a similar role for cubic potentials in canonical form g 3 + g 2 x - 4 x 3 . …
  • Quantum field theory. See Witten (1987).

  • String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).

  • 9: 29.19 Physical Applications
    Bronski et al. (2001) uses Lamé functions in the theory of Bose–Einstein condensates. … Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …
    10: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • W. Wasow (1985) Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.
  • M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.
  • R. L. Wiegel (1960) A presentation of cnoidal wave theory for practical application. J. Fluid Mech. 7 (2), pp. 273–286.
  • E. Witten (1987) Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (4), pp. 525–536.