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Math Help - Holomorphic function on a unbounded domain

  1. #1
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    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
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    i need some help as soon as possible!
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  3. #3
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    Quote Originally Posted by kinkong View Post
    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.
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