# punctured

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## 2 matching pages

##### 1: 1.10 Functions of a Complex Variable
Let $r_{1}=0$, so that the annulus becomes the punctured neighborhood $N$: $0<|z-z_{0}|, and assume that $f(z)$ is analytic in $N$, but not at $z_{0}$. …Lastly, if $a_{n}\not=0$ 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
In a punctured neighborhood $\mathbf{N}$ of a regular singularity $z_{0}$
2.7.4 $w_{j}(z)=(z-z_{0})^{\alpha_{j}}\sum_{s=0}^{\infty}a_{s,j}(z-z_{0})^{s},$ $z\in\mathbf{N}$,
2.7.6 $w_{2}(z)=(z-z_{0})^{\alpha_{2}}\sum_{\begin{subarray}{c}s=0\\ s\neq\alpha_{1}-\alpha_{2}\end{subarray}}^{\infty}b_{s}(z-z_{0})^{s}+cw_{1}(z)% \ln\left(z-z_{0}\right),$ $z\in\mathbf{N}$.