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    Airy functions and applications to physics.

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    • Title:Airy functions and applications to physics.
    •    
    • Author/Creator:ValleĢe, Olivier, 1947-
    • Other Contributors/Collections:Soares, Manuel.
    • Published/Created:Hackensack, NJ : World Scientific, ©2010.

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    • Library of Congress Subjects:Airy functions.
    • Edition:2nd ed.
    • Description:x, 202 p. : ill ; 24 cm.
    • Summary:Addressed mainly to physicist and chemical physicist, this textbook is the result of a broad compilation of current knowledge on analytical properties of Airy functions. In particular, the calculus implying the Airy functions is developed with care. In the latter chapters, examples are given to succinctly illustrate the use of Airy functions in classical and quantum physics. The physicist, for instance in fluid mechanics, can find what he is looking for, in the references for works of molecular physics or in physics of surfaces, and vice versa.
      The knowledge on Airy functions is frequently reviewed. The reason may be found in the need to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. --Book Jacket.
    • Notes:Previous edition: 2004.
      Includes bibliographical references (pages 191-200) and index.
    • ISBN:9781848165489 (hbk.)
      184816548X
    • Contents:1. Historical Introduction: Sir George Biddell Airy
      2. Definitions and Properties
      2.1. Homogeneous Airy functions
      2.1.1. Airy equation
      2.1.2. Elementary properties
      2.1.3. Integral representations
      2.1.4. Ascending and asymptotic series
      2.2. Properties of Airy functions
      2.2.1. Zeros of Airy functions
      2.2.2. spectral zeta function
      2.2.3. Inequalities
      2.2.4. Connection with Bessel functions
      2.2.5. Modulus and phase of Airy functions
      2.3. Inhomogeneous Airy functions
      2.3.1. Definitions
      2.3.2. Properties of inhomogeneous Airy functions
      2.3.3. Ascending series and asymptotic expansion
      2.3.4. Zeros of the Scorer functions
      2.4. Squares and products of Airy functions
      2.4.1. Differential equation and integral representation
      2.4.2. remarkable identity
      2.4.3. product Ai(x)Ai(-x): Airy wavelets
      3. Primitives and Integrals of Airy Functions
      3.1. Primitives containing one Airy function
      3.1.1. In terms of Airy functions
      3.1.2. Ascending series
      3.1.3. Asymptotic expansions
      3.1.4. Primitives of Scorer functions
      3.1.5. Repeated primitives
      3.2. Product of Airy functions
      3.2.1. method of Albright
      3.2.2. Some primitives
      3.3. Other primitives
      3.4. Miscellaneous
      3.5. Elementary integrals
      3.5.1. Particular integrals
      3.5.2. Integrals containing a single Airy function
      3.5.3. Integrals of products of two Airy functions
      3.6. Other integrals
      3.6.1. Integrals involving the Volterra μ-function
      3.6.2. Canonisation of cubic forms
      3.6.3. Integrals with three Airy functions
      3.6.4. Integrals with four Airy functions
      3.6.5. Double integrals
      4. Transformations of Airy Functions
      4.1. Causal properties of Airy functions
      4.1.1. Causal relations
      4.1.2. Green's function of the Airy equation
      4.1.3. Fractional derivatives of Airy functions
      4.2. Airy transform
      4.2.1. Definitions and elementary properties
      4.2.2. Some examples
      4.2.3. Airy polynomials
      4.2.4. particular case: correlation Airy transform
      4.3. Other kinds of transformations
      4.3.1. Laplace transform of Airy functions
      4.3.2. Mellin transform of Airy functions
      4.3.3. Fourier transform of Airy functions
      4.3.4. Hankel transform and the Airy kernel
      4.4. Expansion into Fourier
      -Airy series
      5. Uniform Approximation
      5.1. Oscillating integrals
      5.1.1. method of stationary phase
      5.1.2. uniform approximation of oscillating integrals
      5.1.3. Airy uniform approximation
      5.2. Differential equations of the second order
      5.2.1. JWKB method
      5.2.2. Langer generalisation
      5.3. Inhomogeneous differential equations
      6. Generalisation of Airy Functions
      6.1. Generalisation of the Airy integral
      6.1.1. generalisation of Watson
      6.1.2. Oscillating integrals and catastrophes
      6.2. Third order differential equations
      6.2.1. linear third order differential equation
      6.2.2. Asymptotic solutions
      6.2.3. comparison equation
      6.3. differential equation of the fourth order
      7. Applications to Classical Physics
      7.1. Optics and electromagnetism
      7.2. Fluid mechanics
      7.2.1. Tricomi equation
      7.2.2. Orr-Sommerfeld equation
      7.3. Elasticity
      7.4. heat equation
      7.5. Nonlinear physics
      7.5.1. Korteweg-de Vries equation
      7.5.2. second Painleve equation
      8. Applications to Quantum Physics
      8.1. Schrodinger equation
      8.1.1. Particle in a uniform field
      8.1.2. |x| potential
      8.1.3. Uniform approximation of the Schrodinger equation
      8.2. Evaluation of the Frank-Condon factors
      8.2.1. Franck-Condon principle
      8.2.2. JWKB approximation
      8.2.3. uniform approximation
      8.3. semiclassical Wigner distribution
      8.3.1. Weyl-Wigner formalism
      8.3.2. one-dimensional Wigner distribution
      8.3.3. two-dimensional Wigner distribution
      8.3.4. Configuration of the Wigner distribution
      8.4. Airy transform of the Schrodinger equation
      Appendix A. Numerical Computation of the Airy Functions
      A.1. Homogeneous functions
      A.2. Inhomogeneous functions.
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