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11: Bibliography X
  • H. Xiao, V. Rokhlin, and N. Yarvin (2001) Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems 17 (4), pp. 805–838.
  • 12: Alexander A. Its
     Novokshënov), published by Springer in 1986, Algebro-geometric Approach to Nonlinear Integrable Problems (with E. …
    13: 9.16 Physical Applications
    These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability. … Airy functions play a prominent role in problems defined by nonlinear wave equations. … This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …In the case of the rainbow, the scattering amplitude is expressed in terms of Ai ( x ) , the analysis being similar to that given originally by Airy (1838) for the corresponding problem in optics. An application of the Scorer functions is to the problem of the uniform loading of infinite plates (Rothman (1954b, a)).
    14: 31.17 Physical Applications
    §31.17(i) Addition of Three Quantum Spins
    The problem of adding three quantum spins 𝐬 , 𝐭 , and 𝐮 can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … Consider the following spectral problem on the sphere S 2 : 𝐱 2 = x s 2 + x t 2 + x u 2 = R 2 . … For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). …
    15: 3.7 Ordinary Differential Equations
    §3.7(ii) Taylor-Series Method: Initial-Value Problems
    §3.7(iii) Taylor-Series Method: Boundary-Value Problems
    §3.7(iv) Sturm–Liouville Eigenvalue Problems
    The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system … This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in λ ; compare §3.2(vi). …
    16: Browsers
    Also see About MathML for help for solving problems with MathML in your browser. …
    17: 12.15 Generalized Parabolic Cylinder Functions
    This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …
    18: 12.16 Mathematical Applications
    In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. …
    19: 17.17 Physical Applications
    It involves q -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
    20: 17.18 Methods of Computation
    Shanks (1955) applies such methods in several q -series problems; see Andrews et al. (1986).