Thread: Holomorphic function on a unbounded domain

1. 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

2. i need some help as soon as possible!

3. 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 $\displaystyle e^z$ is bounded there.