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Existence and first properties of transcendental numbers.- Convergent laurent series and formal laurent series.- First results on the values of analytic functions at algebraic points.- Linear differental equations: The lemmas of Shidlovski.- Linear differential equations: A lower bound for the rank of the values of finitely many siegel E-functions at algebraic points.- Linear differential equations: Shidlovski's theorems on the transcendency and algebraic independence of values of siegel E-functions.- Applications of Shidlovski's main theorems to special functions.- Formal power series as solutions of algebraic differential equations.