biconfluent Heun equation
11—20 of 423 matching pages
§31.17 Physical Applications►
§31.17(i) Addition of Three Quantum Spins… ►
§31.17(ii) Other Applications… ►For application of biconfluent Heun functions in a model of an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere see Boyd and Natarov (1998).
§31.13 Asymptotic Approximations… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
§31.4 Solutions Analytic at Two Singularities: Heun Functions… ►To emphasize this property this set of functions is denoted by … ►The eigenvalues satisfy the continued-fraction equation … ►The set depends on the choice of and . ►The solutions (31.4.3) are called the Heun functions. …
§31.6 Path-Multiplicative Solutions►A further extension of the notation (31.4.1) and (31.4.3) is given by ►
15: Gerhard Wolf
… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. … Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. … ►
§31.18 Methods of Computation… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.
§31.3(i) Fuchs–Frobenius Solutions at► denotes the solution of (31.2.1) that corresponds to the exponent at and assumes the value there. … ►
§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities… ►
§31.3(iii) Equivalent Expressions… ►For example, is equal to …
§31.16 Mathematical Applications►
§31.16(i) Uniformization Problem for Heun’s Equation… ►It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. ►
§31.16(ii) Heun Polynomial Products… ►
§31.7(i) Reductions to the Gauss Hypergeometric Function… ►Other reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where . … ►
§31.7(ii) Relations to Lamé Functions… ►equation (31.2.1) becomes Lamé’s equation with independent variable ; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …