Analitičko produljenje funkcije

Olujić, Nediljka (2015) Analitičko produljenje funkcije. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this diploma thesis we proved two important results. Those are the monodromy theorem and the Picard theorem. To prove the monodromy theorem, in chapter ”Regular points and singular points”, we have introduced concepts of regular points of function $f$, singular points of function $f$ and natural boundary of function $f$. We also proved theorem of Ostrowski and theorem of Hadamard. In chapter ”Continuation along curves”, we have introduced concepts of function element $(f, D)$, and we said when the two function elements are direct continuations of each other. We also defined analytic continuation of $(f, D)$ along $\gamma$ and one-parameter family $\gamma_t$ of curves from $\alpha$ to $\beta$ in topological space $X$. From all the above results the proof of the monodromy theorem simply followed and that proof is in chapter called ”The monodromy theorem”. In chapter ”Construction of a modular function” we have observed the set $G$ of all linear fractional transformations $\varphi$ of the form $\varphi(z)=\frac{az+b}{cz+d},$ where $a, b, c, d \in \mathbb{Z}$ and $ad-bc = 1$ Then we introduced a subgroup $\Gamma$ of $G$, and we defined a fundamental domain of $\Gamma$. One of our objectives was the construction of a certain function $\lambda$, which is invariant under $\Gamma$ and which leads to a quick proof of the Picard theorem. Lastly, in chapter called ”The Picard theorem”, we proved that theorem.