Analytic Functions
§Let S Ì C be an open set, and let f(s) be a complex valued function defined on S,   f(s) : S ®  C. Then f(s) is analytic at a point z0 in S if it differentiable at z0 and  also at each point in some neighborhood of z0.
•     It is a fact that if f(s) is analytic at z0 then f has continuous derivatives of all orders at z0. Hence, it has a power series representation at z0.
•     A function f(s) is said to be analytic in S if it has a derivative or is analytic at each point of S.
§Maximum Modulus Theorem: If f(s) is defined and continuous on a closed-bounded set S and analytic on the interior of S, then
•     maxsÎS   êf(s) ê= maxsζS  êf(s) ê
• where ¶S denotes the boundary of S.