# rank of singularity

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## 1—10 of 14 matching pages

##### 1: 31.12 Confluent Forms of Heun’s Equation

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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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##### 2: 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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##### 3: 2.7 Differential Equations

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###### §2.7(ii) Irregular Singularities of Rank 1

… ►Thus a regular singularity has rank 0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. … ►For extensions to singularities of higher rank see Olver and Stenger (1965). … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: …##### 4: 30.2 Differential Equations

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►This equation has regular singularities at $z=\pm 1$ with exponents $\pm \frac{1}{2}\mu $ and an irregular singularity of rank 1 at $z=\mathrm{\infty}$ (if $\gamma \ne 0$).
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##### 5: 2.9 Difference Equations

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►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)).
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##### 6: Bibliography D

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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.
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##### 7: 33.2 Definitions and Basic Properties

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►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)).
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##### 8: 33.14 Definitions and Basic Properties

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►Again, there is a regular singularity at $r=0$ with indices $\mathrm{\ell}+1$ and $-\mathrm{\ell}$, and an irregular singularity of rank 1 at $r=\mathrm{\infty}$.
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##### 9: Bibliography M

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Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank.
Methods Appl. Anal. 4 (3), pp. 250–260.
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##### 10: 13.2 Definitions and Basic Properties

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►This equation has a regular singularity at the origin with indices $0$ and $1-b$, and an irregular singularity at infinity of rank one.
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