Please help me to prove.
Prove that if domain $\displaystyle \Omega$ conformaly maps to unit disc $\displaystyle D=\left \{ \left | z \right |<1 \right \} $then $\displaystyle \Omega$ it's simply connected domain.
Thanks!
Please help me to prove.
Prove that if domain $\displaystyle \Omega$ conformaly maps to unit disc $\displaystyle D=\left \{ \left | z \right |<1 \right \} $then $\displaystyle \Omega$ it's simply connected domain.
Thanks!
You need to assume that the conformal map $\displaystyle f:\Omega\to D$ is bijective (and therefore invertible), otherwise the result is not true.
Suppose that S is a loop in $\displaystyle \Omega$. The image f(S) of S is a loop in the simply-connected space D. So there is a continuous transformation in D that contracts this loop to a point. The image in of that deformation under the map $\displaystyle f^{-1}$ will be a continuous transformation in $\displaystyle \Omega$ which contracts S to a point S to a point.