# annulus

(0.000 seconds)

## 3 matching pages

##### 1: 2.10 Sums and Sequences
Let $f(z)$ be analytic on the annulus $0<|z|, with Laurent expansion
2.10.25 $f(z)=\sum_{n=-\infty}^{\infty}f_{n}z^{n},$ $0<|z|.
where $\mathscr{C}$ is a simple closed contour in the annulus that encloses $z=0$. …
• (c)

The coefficients in the Laurent expansion

2.10.27 $g(z)=\sum_{n=-\infty}^{\infty}g_{n}z^{n},$ $0<|z|,

have known asymptotic behavior as $n\to\pm\infty$.

• 2.10.29 $f_{n}=g_{n}+o\left(r^{-n}\right),$ $n\to\pm\infty$.
##### 2: 1.10 Functions of a Complex Variable
###### §1.10(iii) Laurent Series
Suppose $f(z)$ is analytic in the annulus $r_{1}<|z-z_{0}|, $0\leq r_{1}, and $r\in(r_{1},r_{2})$. Then …The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. 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 $z_{0}$. …
##### 3: 2.7 Differential Equations
these series converging in an annulus $|z|>a$, with at least one of $f_{0}$, $g_{0}$, $g_{1}$ nonzero. …