# irregular solutions

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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: 33.23 Methods of Computation
The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … 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). … This implies decreasing $\ell$ for the regular solutions and increasing $\ell$ for the irregular solutions of §§33.2(iii) and 33.14(iii). …
##### 3: 33.2 Definitions and Basic Properties
###### §33.2(iii) IrregularSolutions$G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$
As in the case of $F_{\ell}\left(\eta,\rho\right)$, the solutions ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$ are analytic functions of $\rho$ when $0<\rho<\infty$. …
##### 6: 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.
• ##### 7: 2.7 Differential Equations
###### §2.7(ii) Irregular Singularities of Rank 1
For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …
##### 8: Bibliography D
• T. M. Dunster (1996c) Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.
• ##### 9: 13.2 Definitions and Basic Properties
This equation has a regular singularity at the origin with indices $0$ and $1-b$, and an irregular singularity at infinity of rank one. …In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\infty$ coalesce into an irregular singularity at $\infty$.
##### 10: 31.12 Confluent Forms of Heun’s Equation
This has one singularity, an irregular singularity of rank $3$ at $z=\infty$. …