asymptotic forms
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1: 31.13 Asymptotic Approximations
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►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).
2: 28.15 Expansions for Small
§28.15 Expansions for Small
…3: 34.8 Approximations for Large Parameters
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►For large values of the parameters in the , , and symbols, different asymptotic forms are obtained depending on which parameters are large.
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4: 28.4 Fourier Series
5: 10.19 Asymptotic Expansions for Large Order
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§10.19(i) Asymptotic Forms
…6: 10.41 Asymptotic Expansions for Large Order
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§10.41(i) Asymptotic Forms
… ►This is because and , do not form an asymptotic scale (§2.1(v)) as ; see Olver (1997b, pp. 422–425). …7: 33.23 Methods of Computation
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►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.
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►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21).
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►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions.
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8: 36.7 Zeros
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►There are also three sets of zero lines in the plane related by rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates is given by
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