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1: 28.2 Definitions and Basic Properties
§28.2(ii) Basic Solutions w I , w II
§28.2(iv) Floquet Solutions
See accompanying text
Figure 28.2.1: Eigenvalues a n ( q ) , b n ( q ) of Mathieu’s equation as functions of q for 0 q 10 , n = 0 , 1 , 2 , 3 , 4 ( a ’s), n = 1 , 2 , 3 , 4 ( b ’s). Magnify
§28.2(vi) Eigenfunctions
For simple roots q of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations …
2: 28.20 Definitions and Basic Properties
28.20.1 w ′′ - ( a - 2 q cosh ( 2 z ) ) w = 0 ,
28.20.2 ( ζ 2 - 1 ) w ′′ + ζ w + ( 4 q ζ 2 - 2 q - a ) w = 0 , ζ = cosh z .
28.20.6 Fe n ( z , q ) = i fe n ( ± i z , q ) , n = 0 , 1 , ,
28.20.7 Ge n ( z , q ) = ge n ( ± i z , q ) , n = 1 , 2 , .
3: 9.16 Physical Applications
A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)). … Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). … 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). …
4: 30.10 Series and Integrals
5: 10.73 Physical Applications
§10.73(ii) Spherical Bessel Functions
6: 17.17 Physical Applications
See Kassel (1995). …
7: 21.9 Integrable Equations
§21.9 Integrable Equations
Typical examples of such equations are the Korteweg--de Vries equation …Here, and in what follows, x , y , and t suffixes indicate partial derivatives. …
8: 28 Mathieu Functions and Hill’s Equation
Chapter 28 Mathieu Functions and Hill’s Equation
9: Simon Ruijsenaars
His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …
  • 10: 10.36 Other Differential Equations
    §10.36 Other Differential Equations