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1: 31.3 Basic Solutions
§31.3 Basic Solutions
§31.3(i) Fuchs–Frobenius Solutions at z = 0
§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
§31.3(iii) Equivalent Expressions
2: 28.29 Definitions and Basic Properties
§28.29(ii) Floquet’s Theorem and the Characteristic Exponent
The basic solutions w I ( z , λ ) , w II ( z , λ ) are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)). …
3: 28.2 Definitions and Basic Properties
§28.2(ii) Basic Solutions w I , w II
(28.2.1) possesses a fundamental pair of solutions w I ( z ; a , q ) , w II ( z ; a , q ) called basic solutions with …
§28.2(vi) Eigenfunctions
For the connection with the basic solutions in §28.2(ii), …
4: 28.4 Fourier Series
For the basic solutions w I and w II see §28.2(ii).
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: 18.38 Mathematical Applications
Differential Equations: Spectral Methods
The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators. …Schneider et al. (2016) discuss DVR/Finite Element solutions of the time-dependent Schrödinger equation. … It has elegant structures, including N -soliton solutions, Lax pairs, and Bäcklund transformations. … has a solution
7: 33.2 Definitions and Basic Properties
§33.2 Definitions and Basic Properties
§33.2(i) Coulomb Wave Equation
§33.2(ii) Regular Solution F ( η , ρ )
§33.2(iii) Irregular Solutions G ( η , ρ ) , H ± ( η , ρ )
As in the case of F ( η , ρ ) , the solutions H ± ( η , ρ ) and G ( η , ρ ) are analytic functions of ρ when 0 < ρ < . …
8: 14.21 Definitions and Basic Properties
§14.21 Definitions and Basic Properties
Standard solutions: the associated Legendre functions P ν μ ( z ) , P ν μ ( z ) , 𝑸 ν μ ( z ) , and 𝑸 ν 1 μ ( z ) . …
§14.21(ii) Numerically Satisfactory Solutions
When ν 1 2 and μ 0 , a numerically satisfactory pair of solutions of (14.21.1) in the half-plane | ph z | 1 2 π is given by P ν μ ( z ) and 𝑸 ν μ ( z ) . …
9: Bibliography C
  • J. Camacho, R. Guimerà, and L. A. N. Amaral (2002) Analytical solution of a model for complex food webs. Phys. Rev. E 65 (3), pp. (030901–1)–(030901–4).
  • L. D. Carr, C. W. Clark, and W. P. Reinhardt (2000) Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.
  • P. A. Clarkson (1991) Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.
  • P. A. Clarkson (2005) Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.
  • M. S. Costa, E. Godoy, R. L. Lamblém, and A. Sri Ranga (2012) Basic hypergeometric functions and orthogonal Laurent polynomials. Proc. Amer. Math. Soc. 140 (6), pp. 2075–2089.
  • 10: 33.14 Definitions and Basic Properties
    §33.14 Definitions and Basic Properties
    §33.14(i) Coulomb Wave Equation
    §33.14(ii) Regular Solution f ( ϵ , ; r )
    §33.14(iii) Irregular Solution h ( ϵ , ; r )
    §33.14(iv) Solutions s ( ϵ , ; r ) and c ( ϵ , ; r )