## Tag Archives: zero

### Some generalizations of theorems of holomorphic function.

$\mathbf{Thm:}$ If $f\not\equiv$ constant is analytic in a domain $U$, then $f(U)$ is also a domain. If $f$ only has poles in $U$, this conclusion still holds.

$\mathbf{Thm:}$ $f(z)$ holomorphic at $z_0\in \overline{\mathbb{C}}$, where it has a zero point of order $k\geq 1$, then there is a neighborhood of $\mathcal{N}(z_0)$ and a neighborhood of $\mathcal{N}(w_0)$ such that every point in $\mathcal{N}(w_0)$ except $w_0$ itself has precisely $k$ distinct inverse images in $\mathcal{N}(z_0)$. If $f$ has pole of order $k$ at $z_0$, conclusion still holds.

$\mathbf{Proof:}$ Suppose $f$ has pole $\phi(z)=\frac{1}{f(z)}$, $\eta=\frac 1 w$. Observe that $w=\frac {1}{\eta}$ carries $|\eta|<\delta$ into the disc $|w|>\frac 1\delta$. If $z_0=\infty$, consider both transformations $\xi=\frac 1z$, $\eta=\frac 1 w$. $\square$

$\mathbf{Corollary:}$ If $w=f(z)$ is univalent(injective) on a domain $G$, then all the zeros and poles of $f(z)$ in $G$  are simple.