About the conformal map $f$ of a slit disc onto a unit disc with the condition $f(i/2)=0$

Question

The slit disc is $D$  $(-1,a]$ where a belongs to $(-1,1)$. The question asks me to find a conformal map from this slit disc to the unit disc and such map satisfy $f(i/2) = 0$. I currently have no certain clue how to construct such a map, but I guess I need to start with certain simpler maps and compose them together to make this $f$. So is there any general method or thought to construct such a map?

reuns · Accepted Answer

With $frac{z-a}{1-az}$ send $Dsetminus(-1,a]$ to $Dsetminus(-1,0]$,

with $i z^{1/2}$ send $Dsetminus(-1,0]$ to $|z|<1,Im(z)>0$,

with the inverse of $zto frac{z-1}{z+1}$ send $|z|<1,Im(z)>0$, to $Re(z)>0,Im(z)>0$,

with $z^2$ send $Re(z)>0,Im(z)>0$ to $Im(z)>0$,

which is biholomorphic to the unit disk with $frac{z-i}{z+i}$.

Composing with an automorphism of the unit disk you get $f(i/2) = 0$

About the conformal map $f$ of a slit disc onto a unit disc with the condition $f(i/2)=0$

One Answer

Add your own answers!

Ask a Question