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1: 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).
2: 31.12 Confluent Forms of Heun’s Equation
Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … This has regular singularities at z = 0 and 1 , and an irregular singularity of rank 1 at z = . … This has irregular singularities at z = 0 and , each of rank 1 . … This has a regular singularity at z = 0 , and an irregular singularity at of rank 2 . … This has one singularity, an irregular singularity of rank 3 at z = . …
3: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1994) Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
  • F. W. J. Olver and F. Stenger (1965) Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.
  • F. W. J. Olver (1965) On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
  • F. W. J. Olver (1997a) Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.
  • 4: 16.21 Differential Equation
    With the classification of §16.8(i), when p < q the only singularities of (16.21.1) are a regular singularity at z = 0 and an irregular singularity at z = . …
    5: 2.7 Differential Equations
    All other singularities are classified as irregular. …
    §2.7(ii) Irregular Singularities of Rank 1
    The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …
    6: 10.72 Mathematical Applications
    These expansions are uniform with respect to z , including the turning point z 0 and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. …
    7: 30.2 Differential Equations
    This equation has regular singularities at z = ± 1 with exponents ± 1 2 μ and an irregular singularity of rank 1 at z = (if γ 0 ). …
    8: 16.8 Differential Equations
    All other singularities are irregular. … Equation (16.8.4) has a regular singularity at z = 0 , and an irregular singularity at z = , whereas (16.8.5) has regular singularities at z = 0 , 1 , and . … Thus in the case p = q the regular singularities of the function on the left-hand side at α and coalesce into an irregular singularity at . …
    9: 10.2 Definitions
    This differential equation has a regular singularity at z = 0 with indices ± ν , and an irregular singularity at z = of rank 1 ; compare §§2.7(i) and 2.7(ii). …
    10: 2.9 Difference Equations
    This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). …