# special function solutions

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## 1—10 of 73 matching pages

##### 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

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##### 3: Bibliography N

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Toda equation and its solutions in special functions.
J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
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##### 4: T. Mark Dunster

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►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.
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##### 5: Howard S. Cohl

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►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.
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##### 6: 31.12 Confluent Forms of Heun’s Equation

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►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation.
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##### 7: Bonita V. Saunders

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►Her research interests include numerical grid generation, numerical solution of partial differential equations, and visualization of special functions.
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##### 8: Frank W. J. Olver

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►, 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.
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##### 9: 32.2 Differential Equations

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►However, for special values of the parameters, equations ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ have special solutions in terms of elementary functions, or special functions defined elsewhere in the DLMF.
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##### 10: 18.38 Mathematical Applications

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►While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations.
However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s.
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