# Holomorphic function on a unbounded domain

• Apr 4th 2011, 04:30 PM
kinkong
Holomorphic function on a unbounded domain
I have problem in understanging the following question
Q. Show by means of an example that a non-constant holomorphic function on a n unbounded domain need not achieve it's maximum modulus on the boundary of that domain

Answer: here we assume that tha boundary is not empty.
Maximum Modulus Principle says that
"|f(z)| can only achieve its maximum value on the boundary unless it is constant"
So i am wondering how can i connect the MMP with the above question.
Any help will be appreciated....Thanks
• Apr 5th 2011, 12:01 AM
kinkong
i need some help as soon as possible!
• Apr 5th 2011, 12:46 AM
Opalg
Quote:

Originally Posted by kinkong
I have problem in understanging the following question
Q. Show by means of an example that a non-constant holomorphic function on a n unbounded domain need not achieve it's maximum modulus on the boundary of that domain

Answer: here we assume that tha boundary is not empty.
Maximum Modulus Principle says that
"|f(z)| can only achieve its maximum value on the boundary unless it is constant"
So i am wondering how can i connect the MMP with the above question.
Any help will be appreciated....Thanks

Suppose you take the domain to be the right-hand half-plane. The boundary is then the imaginary axis, and the function $e^z$ is bounded there.