# punctured

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## 2 matching pages

##### 1: 1.10 Functions of a Complex Variable

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►Let ${r}_{1}=0$, so that the annulus becomes the

*punctured neighborhood*$N$: $$, and assume that $f(z)$ is analytic in $N$, but not at ${z}_{0}$. …Lastly, if ${a}_{n}\ne 0$ for infinitely many negative $n$, then ${z}_{0}$ is an*isolated essential singularity*. … ►Suppose $F(z)$ is multivalued and $a$ is a point such that there exists a branch of $F(z)$ in a cut neighborhood of $a$, but there does not exist a branch of $F(z)$ in any punctured neighborhood of $a$. …##### 2: 2.7 Differential Equations

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►In a punctured neighborhood $\mathbf{N}$ of a regular singularity ${z}_{0}$
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►

2.7.4
$${w}_{j}(z)={(z-{z}_{0})}^{{\alpha}_{j}}\sum _{s=0}^{\mathrm{\infty}}{a}_{s,j}{(z-{z}_{0})}^{s},$$
$z\in \mathbf{N}$,

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2.7.6
$${w}_{2}(z)={(z-{z}_{0})}^{{\alpha}_{2}}\sum _{\begin{array}{c}\hfill s=0\hfill \\ \hfill s\ne {\alpha}_{1}-{\alpha}_{2}\hfill \end{array}}^{\mathrm{\infty}}{b}_{s}{(z-{z}_{0})}^{s}+c{w}_{1}(z)\mathrm{ln}\left(z-{z}_{0}\right),$$
$z\in \mathbf{N}$.

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