Karl Westergren: Reellt analytiska funktioner av en variabel

Abstract: In this thesis, we discuss real analytic functions of one variable, including their basic properties and illustrative examples. All real analytic functions are in particular smooth. However, not all smooth functions are real analytic, since analyticity is a stronger condition which ensures that the function is locally representable by an absolutely convergent power series. Within the class of real analytic functions, the functions themselves are, locally around a point, uniquely determined by their derivatives of all orders at that point. This property makes them rigid and can, for example, be useful in the theory of ordinary differential equations.