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 z**0* in *S* if it differentiable at *z*0 and also at each point in some neighborhood of *z*0.

It is a fact
that if *f*(s) is analytic at *z*0 then *f *has continuous derivatives
of all
orders at *z*0. Hence, it has a power series representation at *z*0.

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

maxsÎ*S * ê*f*(s) ê= maxsÎ¶*S * ê*f*(s) ê

where ¶*S *denotes the boundary of *S*.