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special function solutions

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1: 32.10 Special Function Solutions
§32.10 Special Function Solutions
§32.10(ii) Second Painlevé Equation
§32.10(iii) Third Painlevé Equation
§32.10(iv) Fourth Painlevé Equation
2: 15.17 Mathematical Applications
3: Bibliography N
  • A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
  • 4: T. Mark Dunster
    He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …
    5: Howard S. Cohl
    Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and q -series. …
    6: 31.12 Confluent Forms of Heun’s Equation
    Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions30.12) are special cases of solutions of the confluent Heun equation. …
    7: Bonita V. Saunders
    Her research interests include numerical grid generation, numerical solution of partial differential equations, and visualization of special functions. …
    8: Frank W. J. Olver
    , the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …
    9: 32.2 Differential Equations
    However, for special values of the parameters, equations P II P VI  have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF. …
    10: 31.1 Special Notation
    §31.1 Special Notation
    (For other notation see Notation for the Special Functions.)
    x , y real variables.
    The main functions treated in this chapter are H ( a , q ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) 𝐻𝑓 m ( a , q m ; α , β , γ , δ ; z ) , ( s 1 , s 2 ) 𝐻𝑓 m ν ( a , q m ; α , β , γ , δ ; z ) , and the polynomial 𝐻𝑝 n , m ( a , q n , m ; n , β , γ , δ ; z ) . …Sometimes the parameters are suppressed.