Analytic Functions

In this chapter we introduce the definition of a k-valued analytic function on an open set $$\Omega $$ in $$k^n$$ where k is a local field k and prove the main results about them. When $$k\simeq \mathbb {C}$$ this is a familiar concept defined via the existence of the first derivative with respect to a complex variable. When $$k=\mathbb {R}$$ one may define a $$\mathbb {R}$$ -valued analytic function on $$\Omega $$ as the restriction to $$\Omega $$ of a $$\mathbb {C}$$ -valued analytic function on an open set $$\tilde{\Omega }$$ in $$\mathbb {C}^n$$ containing $$\Omega $$ . An alternative approach (following Weierstrass) is via convergent power series over $$\mathbb {R}$$ . When k is not archimedean, only the second approach is available and is adopted in this chapter.