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.
