Power Series Extensions of Certain Functions of a Real Variable

Abstract

The domain of the function $ f(x)=\cos \sqrt{x} $ is the set of all nonnegative real numbers. In this article, we will show how to use power series to extend this function to an analytic function defined on the entire real line. We will then show how this analytic extension of $ f(x) $ makes it easier and quicker for calculus students to compute derivatives of $ f(x) $ at the origin. We will moreover describe the process of extending the domain of any function of the form $ g(\sqrt{x}) $ for a given even analytic function $ g(x) $.