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oscillations of chains


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1: 10.73 Physical Applications
§10.73(i) Bessel and Modified Bessel Functions
Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … In the theory of plates and shells, the oscillations of a circular plate are determined by the differential equation
10.73.3 4 W + λ 2 2 W t 2 = 0 .
2: Sidebar 9.SB2: Interference Patterns in Caustics
The oscillating intensity of the interference fringes across the caustic is described by the Airy function.
3: 17.17 Physical Applications
See Kassel (1995). … It involves q -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
4: 6.17 Physical Applications
Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.
5: 17.12 Bailey Pairs
When (17.12.5) is iterated the resulting infinite sequence of Bailey pairs is called a Bailey Chain. … The Bailey pair and Bailey chain concepts have been extended considerably. …
6: 8.24 Physical Applications
The function γ ( a , x ) appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …
7: 18.39 Physical Applications
For a harmonic oscillator, the potential energy is given by …
8: 7.21 Physical Applications
Fried and Conte (1961) mentions the role of w ( z ) in the theory of linearized waves or oscillations in a hot plasma; w ( z ) is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …
9: 22.19 Physical Applications
§22.19(ii) Classical Dynamics: The Quartic Oscillator
For an initial displacement with 1 / β | a | < 2 / β , bounded oscillations take place near one of the two points of stable equilibrium x = ± 1 / β . Such oscillations, of period 2 K ( k ) / η , with modulus k = 1 / 2 η 1 are given by: …
10: Bibliography
  • V. È. Adler (1994) Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.
  • G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
  • G. E. Andrews (2001) Bailey’s Transform, Lemma, Chains and Tree. In Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.