Lukivska D. The properties of generalized loxodromic and generalized elliptic functions

The PhD Thesis is devoted to a generalization of loxodromic and elliptic functions. We introduce for the first time the notion of p-loxodromic function; the criterion of p-loxodromicity is proved; the representation of holomorphic p-loxodromic function is found; the theorem about the number of zeros and poles of p-loxodromic function is proved; a logarithmic spiral, containing zeros and poles of p-loxodromic function is found; a Julia exceptionality of p-loxodromic functions is established. The PhD Thesis also contains another generalization of loxodromicity, namely rationally-loxodromic functions. Also, the notion of quasi-elliptic function is introduced. For the class of quasi-elliptic function analogues of the certainclassic Weierstrass functions are constructed. The connection between quasi-elliptic and p-loxodromic functions is obtained. Additionally, the PhD Thesis also contains alternative generalizations of loxodromic and elliptic functions, so called modulo-loxodromic and modulo-elliptic functions. The theorems, which describe relations between the classes of modulo-loxodromic and p-loxodromic as well as between modulo-elliptic and quasi-elliptiс functions are proved.