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11: 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.
  • 12: 22.19 Physical Applications
    where V ( x ) is the potential energy, and x ( t ) is the coordinate as a function of time t . The potential
    22.19.5 V ( x ) = ± 1 2 x 2 ± 1 4 β x 4
    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)). Its dynamics for purely imaginary time is connected to the theory of instantons (Itzykson and Zuber (1980, p. 572), Schäfer and Shuryak (1998)), to WKB theory, and to large-order perturbation theory (Bender and Wu (1973), Simon (1982)). …
    13: 9.16 Physical Applications
    Details of the Airy theory are given in van de Hulst (1957) in the chapter on the optics of a raindrop. … Extensive use is made of Airy functions in investigations in the theory of electromagnetic diffraction and radiowave propagation (Fock (1965)). … Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. … Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …
    14: Bibliography G
  • F. G. Garvan and M. E. H. Ismail (Eds.) (2001) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.
  • J. S. Geronimo, O. Bruno, and W. Van Assche (2004) WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.
  • R. G. Gordon (1970) Constructing wavefunctions for nonlocal potentials. J. Chem. Phys. 52, pp. 6211–6217.
  • D. Gottlieb and S. A. Orszag (1977) Numerical Analysis of Spectral Methods: Theory and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA.
  • J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.
  • 15: 5.20 Physical Applications
    Suppose the potential energy of a gas of n point charges with positions x 1 , x 2 , , x n and free to move on the infinite line - < x < , is given by
    5.20.1 W = 1 2 = 1 n x 2 - 1 < j n ln | x - x j | .
    5.20.2 P ( x 1 , , x n ) = C exp ( - W / ( k T ) ) ,
    Elementary Particles
    Veneziano (1968) identifies relationships between particle scattering amplitudes described by the beta function, an important early development in string theory. …
    16: Bibliography R
  • H. Rademacher (1973) Topics in Analytic Number Theory. Springer-Verlag, New York.
  • B. Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inauguraldissertation, Göttingen.
  • S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.
  • K. H. Rosen (2004) Elementary Number Theory and its Applications. 5th edition, Addison-Wesley, Reading, MA.
  • G. Rota (1964) On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pp. 340–368.
  • 17: 33.22 Particle Scattering and Atomic and Molecular Spectra
    At positive energies E > 0 , ρ 0 , and: … R = m e c α 2 / ( 2 ) . … Both variable sets may be used for attractive and repulsive potentials: the ( ϵ , r ) set cannot be used for a zero potential because this would imply r = 0 for all s , and the ( η , ρ ) set cannot be used for zero energy E because this would imply ρ = 0 always. …
    §33.22(vi) Solutions Inside the Turning Point
    18: Bibliography H
  • R. L. Hall, N. Saad, and K. D. Sen (2010) Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.
  • G. H. Hardy and E. M. Wright (1979) An Introduction to the Theory of Numbers. 5th edition, The Clarendon Press Oxford University Press, New York-Oxford.
  • D. R. Herrick and S. O’Connor (1998) Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.
  • E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
  • I. Huang and S. Huang (1999) Bernoulli numbers and polynomials via residues. J. Number Theory 76 (2), pp. 178–193.
  • 19: Bibliography N
  • W. Narkiewicz (2000) The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.
  • J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
  • N. Nielsen (1906a) Handbuch der Theorie der Gammafunktion. B. G. Teubner, Leipzig (German).
  • N. Nielsen (1965) Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten. Chelsea Publishing Co., New York (German).
  • Number Theory Web (website)
  • 20: Bibliography
  • M. M. Agrest and M. S. Maksimov (1971) Theory of Incomplete Cylindrical Functions and Their Applications. Springer-Verlag, Berlin.
  • L. V. Ahlfors (1966) Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable. 2nd edition, McGraw-Hill Book Co., New York.
  • A. R. Ahmadi and S. E. Widnall (1985) Unsteady lifting-line theory as a singular-perturbation problem. J. Fluid Mech 153, pp. 59–81.
  • H. H. Aly, H. J. W. Müller-Kirsten, and N. Vahedi-Faridi (1975) Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.
  • T. M. Apostol and I. Niven (1994) Number Theory. In The New Encyclopaedia Britannica, Vol. 25, pp. 14–37.