# solutions near irregular singularities

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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: 16.8 Differential Equations

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►All other singularities are

*irregular*. … ►Equation (16.8.4) has a regular singularity at $z=0$, and an irregular singularity at $z=\mathrm{\infty}$, whereas (16.8.5) has regular singularities at $z=0$, $1$, and $\mathrm{\infty}$. … ►When no ${b}_{j}$ is an integer, and no two ${b}_{j}$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by … ►In this reference it is also explained that in general when $q>1$ no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near $z=1$. … ►Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha $ and $\mathrm{\infty}$ coalesce into an irregular singularity at $\mathrm{\infty}$. …##### 3: 31.12 Confluent Forms of Heun’s Equation

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►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.
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►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty}$.
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►This has irregular singularities at $z=0$ and $\mathrm{\infty}$, each of rank $1$.
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►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty}$ of rank $2$.
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►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty}$.
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##### 4: 31.18 Methods of Computation

###### §31.18 Methods of Computation

►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of $z$; see Laĭ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …##### 5: 28.2 Definitions and Basic Properties

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►This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\mathrm{\infty}$.
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►Since (28.2.1) has no finite singularities its solutions are entire functions of $z$.
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###### §28.2(iv) Floquet Solutions

… ►The Fourier series of a Floquet solution …leads to a Floquet solution. …##### 6: 15.19 Methods of Computation

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►As noted in §3.7(ii), the integration path should be chosen so that the wanted solution grows in magnitude at least as fast as all other solutions.
However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases.
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##### 7: Bibliography O

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Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one.
Proc. Roy. Soc. London Ser. A 445, pp. 39–56.
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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one.
Proc. Roy. Soc. London Ser. A 454, pp. 1–29.
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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.
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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.
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Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity.
Methods Appl. Anal. 4 (4), pp. 375–403.
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##### 8: 33.2 Definitions and Basic Properties

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###### §33.2(i) Coulomb Wave Equation

… ►This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell}+1$ and $-\mathrm{\ell}$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty}$ (§§2.7(i), 2.7(ii)). … ►###### §33.2(ii) Regular Solution ${F}_{\mathrm{\ell}}(\eta ,\rho )$

… ►###### §33.2(iii) Irregular Solutions ${G}_{\mathrm{\ell}}(\eta ,\rho ),{H}_{\mathrm{\ell}}^{\pm}(\eta ,\rho )$

… ►As in the case of ${F}_{\mathrm{\ell}}(\eta ,\rho )$, the solutions ${H}_{\mathrm{\ell}}^{\pm}(\eta ,\rho )$ and ${G}_{\mathrm{\ell}}(\eta ,\rho )$ are analytic functions of $\rho $ when $$. …##### 9: 2.7 Differential Equations

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►All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.
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►All other singularities are classified as

*irregular*. … ►###### §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: … ►In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere. …##### 10: Bibliography B

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Zero-field susceptibility of the two-dimensional Ising model near
${T}_{c}$
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Phys. Rev. Lett. 31, pp. 1409–1411.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Uniform asymptotic expansions of integrals with stationary point near algebraic singularity.
Comm. Pure Appl. Math. 19, pp. 353–370.
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The fifty-two icosahedral solutions to Painlevé VI.
J. Reine Angew. Math. 596, pp. 183–214.
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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions.
SIAM J. Math. Anal. 17 (2), pp. 422–450.
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