# Find a conformal map f(z) which maps the unit disc D bijectively onto the whole complex plane minus the non-negative real axis

Q: Find a conformal map f(z) which maps which maps the unit disc D bijectively onto the whole complex plane minus the non-negative real axis, and which in addition satisfies $\lim_{z\to -1}f(z)=0$ and $f(0)=-7$.

My Attempt:

I know that $\frac{z+1}{z-1}$ goes to the left half of imaginary axis.

Now, I take the expression raised by 2, $\frac{(z+1)^2}{(z-1)^2}$, and get the whole complex-plane except the non-negative real axis.

Am I in the right direction? And what must I do to satisfy

$\lim_{z\to -1}f(z)=0$ and $f(0)=-7$?

• It goes to 0. But how about $f(0)=-7$? – Aerdennis Mar 9 '17 at 18:05
• $z \rightarrow i \cdot \frac{1+z}{1-z} \rightarrow \left ( i \cdot \frac{1+z}{1-z} \right )^2=-\left (\frac{1+z}{1-z} \right )^2\rightarrow -7\left (\frac{1+z}{1-z} \right )^2$ – Daniil Mar 9 '17 at 18:41
• Why did you rotate it by pi? Is it because it fullfill f(0)=-7? – Aerdennis Mar 10 '17 at 9:50
• 1) map unit disk to the upper half-plane 2) map upper half-plane to the whole plane minus nonnegative real axis 3) is made in order to satisfy initial condition – Daniil Mar 11 '17 at 22:11