Katsala R. The expansion of functions in continued fraction.

The thesis is devoted to the investigation of approaching methods for one real variable functions by continued fractions based on inverse derivatives and second-type inverse derivatives (the approaching by Thiele continued fraction and Thiele-like quasi-inverse continued fraction). New properties of reciprocal derivatives are obtained. An equivalence of two approaches of regular continued -fraction building are proven and correspondence of Thiele continued fraction to power series is shown. Formulas for expressed of the second-type inverse derivatives between the derivatives by means of two Hankel's determinants quotient are obtained. Evaluation between second-type inverse derivatives and inverse derivatives of Thiele is shown. A connections between Tiele continued fraction and Tiele-like quasi-inverse continued fraction are established, as well as a interconnection between these fractions approximants and Padй approximants are obtained. The new expansions in continued fractions of some one real variable functions are obtained. Convergence conditions of Tiele continued fraction and Tiele-like quasi-inverse continued fraction are received, the convergence region of the constructed expansions are established. Calculable properties of got expansions are practically tested.