# rank of singularity

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## 11—14 of 14 matching pages

##### 11: 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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##### 12: 13.14 Definitions and Basic Properties

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►It has a regular singularity at the origin with indices $\frac{1}{2}\pm \mu $, and an irregular singularity at infinity of rank one.
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##### 13: 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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##### 14: 15.17 Mathematical Applications

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►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,\mathbb{R})$, and spherical functions on certain nonsymmetric Gelfand pairs.
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►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure.
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