(Ebook) Airy Functions and Applications to Physics 2nd Edition by Olivier Vallee, Manuel Soares ISBN 184816548X 9781848165489

(Ebook) Airy Functions and Applications to Physics 2nd Edition by Olivier Vallee, Manuel Soares ISBN 184816548X 9781848165489

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ISBN 10: 184816548X 
ISBN 13: 9781848165489
Author: Olivier Vallee, Manuel Soares

(Ebook) Airy Functions and Applications to Physics 2nd Table of contents:

1. A Historical Introduction: Sir George Biddell Airy
2. Definitions and Properties
2.1 Homogeneous Airy functions
2.1.1 The Airy equation
2.1.2 Elementary properties
2.1.2.1 Wronskians of homogeneous Airy functions
2.1.2.2 Particular values of Airy functions
2.1.2.3 Relations between Airy functions
2.1.3 Integral representations
2.1.4 Ascending and asymptotic series
2.1.4.1 Expansion of Ai near the origin
2.1.4.2 Ascending series of Ai and Bi
2.1.4.3 Asymptotic expansion of Ai and Bi
2.1.4.4 The Stokes phenomenon
2.2 Properties of Airy functions
2.2.1 Zeros of Airy functions
2.2.2 The spectral zeta function
2.2.3 Inequalities
2.2.4 Connection with Bessel functions
2.2.5 Modulus and phase of Airy functions
2.2.5.1 Definitions
2.2.5.2 Differential equations
2.2.5.3 Asymptotic expansions
2.2.5.4 Functions of positive arguments
2.3 Inhomogeneous Airy functions
2.3.1 Definitions
2.3.2 Properties of inhomogeneous Airy functions
2.3.2.1 Values at the origin
2.3.2.2 Other integral representations
2.3.3 Ascending series and asymptotic expansion
2.3.3.1 Ascending series
2.3.3.2 Asymptotic expansions
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 A remarkable identity
2.4.3 The 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 The 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
3.5.2.1 Integrals involving algebraic functions
3.5.2.2 Integrals involving transcendental functions
3.5.3 Integrals of products of two Airy functions
3.6 Other integrals
3.6.1 Integrals involving the Volterra -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 The Airy transform
4.2.1 Definitions and elementary properties
4.2.2 Some examples
4.2.3 Airy polynomials
4.2.4 A particular case: correlation Airy transform
4.2.4.1 Properties of the correlation Airy transform
4.2.4.2 Some examples
4.2.4.3 Self-reciprocal transforms
4.2.4.4 Application to the Airy kernel
4.2.4.5 The cat of Hermite functions
4.2.4.6 Airy averaging
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
Exercises
5. The Uniform Approximation
5.1 Oscillating integrals
5.1.1 The method of stationary phase
5.1.2 The uniform approximation of oscillating integrals
5.1.3 The Airy uniform approximation
5.2 Differential equations of the second order
5.2.1 The JWKB method
5.2.2 The Langer generalisation
5.3 Inhomogeneous differential equations
Exercises
6. Generalisation of Airy Functions
6.1 Generalisation of the Airy integral
6.1.1 The generalisation of Watson
6.1.2 Oscillating integrals and catastrophes
6.2 Third order differential equations
6.2.1 The linear third order differential equation
6.2.2 Asymptotic solutions
6.2.3 The comparison equation
6.3 A differential equation of the fourth order
Exercises
7. Applications to Classical Physics
7.1 Optics and electromagnetism
7.2 Fluid mechanics
7.2.1 The Tricomi equation
7.2.2 The Orr{Sommerfeld equation
7.2.2.1 Plane flow of an incompressible viscous fluid
7.2.2.2 Stability of an almost parallel ow
7.3 Elasticity
7.4 The heat equation
7.5 Nonlinear physics
7.5.1 Korteweg–de Vries equation
7.5.1.1 The linearised Korteweg–de Vries equation
7.5.1.2 Similarity solutions
7.5.2 The second Painleve equation
7.5.2.1 The Painleve equations
7.5.2.2 An integral equation
7.5.2.3 Rational solutions
Exercises
8. Applications to Quantum Physics
8.1 The Schrodinger equation
8.1.1 Particle in a uniform field
8.1.1.1 The stationary case
8.1.1.2 The time dependent case
8.1.2 The |x| potential
8.1.3 Uniform approximation of the Schrodinger equation
8.1.3.1 The JWKB approximation
8.1.3.2 The Airy uniform approximation
8.1.3.3 Exact vs approximate wave functions
8.2 Evaluation of the Franck–Condon factors
8.2.1 The Franck–Condon principle
8.2.2 The JWKB approximation
8.2.3 The uniform approximation
8.3 The semiclassical Wigner distribution
8.3.1 The Weyl–Wigner formalism
8.3.2 The one-dimensional Wigner distribution
8.3.3 The two-dimensional Wigner distribution
8.3.4 Configuration of the Wigner distribution
8.4 Airy transform of the Schrodinger equation

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Tags: Olivier Vallee, Manuel Soares, Functions, Physics