Complex analysis (conformal?) mapping question probably easy

[AxiomOfChoice]

Homework Statement

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

Homework Equations

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 $(-\pi/4, \pi/4)$.

 

[tiny-tim]

Hi AxiomOfChoice!

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.

why??

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

 

[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.