Power Series

Abstract

AbstractThis chapter is devoted to an important tool for constructing holomorphic functions: convergent power series. It is the basis for the introduction of new non-algebraic holomorphic functions, the elementary transcendental functions. It turns out that power series play an even more central role in the theory of holomorphic functions, a role beyond enabling the construction of complex transcendental functions that are the extension of the real transcendental functions. A much stronger result holds. All holomorphic functions are (at least locally) convergent power series. This will be proven in the next chapter.KeywordsPower SeriesExponential FunctionHolomorphic FunctionAnalytic ContinuationMeromorphic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.