- #1
librastar
- 15
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Homework Statement
Find the maximum of |ez| on the closed unit disc.
Homework Equations
|ez| is the modulus of ez
z belongs to complex plane
Maximum Madulus Theorem - Let G be a bounded open set in complex plane and suppose f is a continuous function on G closure which is analytic in G. Then max{|f(z)|: z in G closure} = max{|f(z)|: z in boundary of G}
The Attempt at a Solution
At first glance I was thinking about using Maximum Modulus Theorem by setting f(z) = ez and let G be a open unit disc. Clearly f(z) is analytic in G and is continuous on G closure (the closed unit disc), so I should be able to apply MMT to conclude that the modulus assumes maximum on the boundary.
However I got confused when determining how do I describe |f(z)| on the boundary, in another word, how do I determine what value of z to use?
Please let me know if my approach is right and perhaps give me some advice or hint.
I appreciate all the helps you provide.