# Konformalna preslikavanja

Pehar, Antonija (2016) Konformalna preslikavanja. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

The most important result of this diploma thesis is the Riemann mapping theorem. Before we could prove it, we defined the conformal mappings in the first chapter. After that we studied linear fractional transformations and their connection with those mappings. The important part of proving the Riemann mapping theorem are normal families which we defined in the third chapter. We also proved that if $F \subset H(\Omega)$ and $F$ is uniformly bounded on every compact subset of the region $\Omega$; then $F$ is a normal family. After we defined conformally equivalent regions, we could prove the Riemann mapping theorem. In the fifth chapter we defined the class $S$ ; which is the class of all $f \in H(U)$ which are one-to-one in $U$ and satisfy $f(0)=0, f'(0)=1$, and proved some of its properties. In the chapter "Continuity at the boundary" we defined simple boundary points and proved the next statement: if $\Omega$ is a bounded simply connected region in the plane and if every boundary point of $\Omega$ is simple, then every conformal mapping of $\Omega$ onto $U$ extends to a homeomorphism off $\Omega$ onto $U$. In the last chapter we studied the conformal mapping of an annulus and proved this interesting property: $A(r_1, R_1)$ and $A(r_2, R_2)$ are conformally equivalent if and only if $R_1/r_1=R_2/r_2$.