The logarithmic conjugation theorem and conformal equivalence between multiply connected domains - Löfström,Martin (Bachelor’s Theses in Mathematical Sciences, 2022)
The existence of a primitive of analytic functions, of an analytic logarithm of zero-free analytic functions, and of a harmonic conjugate to any harmonic function are fundamental propositions of elementary complex analysis, holding on simply-connected domains. We find that with suitable corrections, analogous statements can be made on multiply connected domains, foremost of these is these being the logarithmic conjugation theorem. This latter theorem is used to prove that isolated singularities on Hardy spaces are removable, as well as to prove the doubly connected mapping theorem. For more general multiply connected domains, we will prove conformal equivalence with slit domains, in order to lay the foundation to the multipy connected mapping theorem.