Does the Cauchy Integral Formula by itself proves that the Dirichlet Problem is solvable for any simply connected region?

I was looking for the proof of the Poisson Integral Formula and one uses the fact that if $u$ is harmonic in the closed disk $\bar D$ then (because the disk is simply connected) there is $f=u+iv$ holomorphic in $\bar D$, and we have an explicit formula for $f$ by knowing its value on $\partial D$ and using Cauchy Integral Formula. By manipulating these results for $f$, we can take back $u=\operatorname{Re}(f)$ and that is Poisson Integral Formula. In another result, by given a function $f$ in $\partial D$ and by defining a function $u$ as the Poisson Integral in the interior of the disk and as $f$ on the boundary, $u$ solves the Dirichlet Problem, and then the problem is solvable in the unit disk. As said in "Function Theory of One Complex Variable" by Greene "the Cauchy integral formula not only reproduces holomorphic functions but also creates them".

My question is, does this method works for an arbitrary simply connected region $D$? Because the passage from harmonic to holomorphic function uses only the fact of the region being simply connected. Of course we will not get an explicit formula, but by defining $u$ as the real part of the Cauchy Integral Formula, and $f$ on the boundary, will the Dirichlet Problem be solvable in $D$?



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