# oscillation of chains

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## 1—10 of 28 matching pages

##### 1: 10.73 Physical Applications

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###### §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
$${\nabla}^{4}W+{\lambda}^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$$

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##### 2: Sidebar 9.SB2: Interference Patterns in Caustics

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►The oscillating intensity of the interference fringes across the caustic is described by the Airy function.

##### 3: 17.17 Physical Applications

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►See Kassel (1995).
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►It involves $q$-generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials.
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##### 4: 6.17 Physical Applications

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►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

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►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

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►The function $\gamma (a,x)$ 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)).
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##### 7: 18.39 Physical Applications

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►For a harmonic oscillator, the potential energy is given by
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##### 8: 7.21 Physical Applications

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►Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the

*plasma dispersion function*or*Faddeeva (or Faddeyeva) function*; see Faddeeva and Terent’ev (1954). …##### 9: 22.19 Physical Applications

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###### §22.19(ii) Classical Dynamics: The Quartic Oscillator

… ►For an initial displacement with $$, bounded oscillations take place near one of the two points of stable equilibrium $x=\pm \sqrt{1/\beta}$. Such oscillations, of period $2K\left(k\right)/\sqrt{\eta}$, with modulus $k=1/\sqrt{2-{\eta}^{-1}}$ are given by: …##### 10: Bibliography

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Nonlinear chains and Painlevé equations.
Phys. D 73 (4), pp. 335–351.
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Umbral calculus, Bailey chains, and pentagonal number theorems.
J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
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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.
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