# Homework Help: Complex analysis (conformal?) mapping question probably easy

1. Sep 17, 2009

### AxiomOfChoice

1. The problem statement, all variables and given/known data
We're supposed to find a bijective mapping from the open unit disk $\{z : |z| < 1\}$ to the sector $\{z: z = re^{i \theta}, r > 0, -\pi/4 < \theta < \pi/4 \}$.

2. Relevant equations

3. The attempt at a solution
This is confusing me. I tried to find a function that would map $[0,1)$, which is the set of possible values of $r$ in the domain, injectively onto $(0,\infty)$, which is the set of possible values of $r$ in the range. The best thing I could come up with is $f(r) = \dfrac{1}{r(1-r)} - 4$, but this is clearly not one-to-one, and it hits zero. What's more, I'm not sure how to find a function that will map the possible values for $\text{Arg }z$, which are $-\pi < \text{Arg }z \leq \pi$, injectively onto $(-\pi/4, \pi/4)$.

2. Sep 17, 2009

### tiny-tim

Hi AxiomOfChoice!
why??

Hint: get the boundary right, and everything else should fit in.

3. Sep 17, 2009

### Dick

Don't try and mess around individually with r and theta. Just think about analytic functions. For example 1/(1-z) maps the disk into a half plane, right? Now find another function that can take a half plane into a wedge. Put them together.