# irregular singularities of rank 1

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## 1—10 of 12 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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##### 2: 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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##### 3: 2.7 Differential Equations

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###### §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. …##### 4: 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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##### 5: 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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##### 6: 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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##### 7: 10.2 Definitions

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►This differential equation has a regular singularity at $z=0$ with indices $\pm \nu $, and an irregular singularity at $z=\mathrm{\infty}$ of rank
$1$; compare §§2.7(i) and 2.7(ii).
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##### 8: 10.47 Definitions and Basic Properties

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►Equations (10.47.1) and (10.47.2) each have a regular singularity at $z=0$ with indices $n$, $-n-1$, and an irregular singularity at $z=\mathrm{\infty}$ of rank
$1$; compare §§2.7(i)–2.7(ii).
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##### 9: 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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##### 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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