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Math Help - Find all entire functions f

  1. #1
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    Find all entire functions f

    Find all the entire functions f such that
    f(0) = i and \left |f(z)- \cos z  \right |\geq{\sqrt{2}}
    for all {z\in{\mathbb{C}}}.


    My attempt at this question...
    Let g(z) = \frac{1}{f(z)-\cos z}.
    Since f(z) and \cos z are entre and f(z) - \cos z \neq 0, \forall {z\in{\mathbb{C}}} (since \left |f(z)- cos z  \right |\geq{\sqrt{2}}).
    \Rightarrow g(z) is entire & \left |g(z) \right |= \frac{1}{\left |f(z)-\cos z\right |} \leq \frac{1}{\sqrt{2}}, \forall {z\in{\mathbb{C}}}.
    By Liouville's Theorem, g(z) \equiv k for some {k\in{\mathbb{C}}.
    \Rightarrow \frac{1}{f(z)-\cos z} \equiv k for some {k\in{\mathbb{C}}, k \neq 0, since f(z) - \cos z \neq 0, \forall {z\in{\mathbb{C}}}.
    Hence f(z) = \frac{1}{k} + \cos z
    Now it is given that f(0) = i, and cos(0) = 1, hence \frac{1}{k} = i-1.
    \Rightarrow f(z) = i - 1 + \cos z

    Is this correct? Are there any other solutions? Thanks in advance!.
    Last edited by alphabeta89; October 17th 2012 at 12:44 AM.
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  2. #2
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    Re: Find all entire functions f

    Quote Originally Posted by alphabeta89 View Post
    Find all the entire functions f such that
    f(0) = i and \left |f(z)- \cos z  \right |\geq{2}
    for all {z\in{\mathbb{C}}}.
    \text{Umm... } f(0) = i \text{ and } \cos(0) = \frac{e^{i(0)} + e^{-i(0)}}{2} = \frac{1 + 1}{2} = 1,

    \text{so } 2 \le \lVert f(0)- \cos(0) \rVert = \lVert -1 + i \rVert = \sqrt{2} < 2.

    The answer to this problem is that there are no such functions.
    Last edited by johnsomeone; October 16th 2012 at 11:35 PM.
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  3. #3
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    Re: Find all entire functions f

    Quote Originally Posted by johnsomeone View Post
    \text{Umm... } f(0) = i \text{ and } \cos(0) = \frac{e^{i(0)} + e^{-i(0)}}{2} = \frac{1 + 1}{2} = 1,

    \text{so } 2 \le \lVert f(0)- \cos(0) \rVert = \lVert -1 + i \rVert = \sqrt{2} < 2.

    The answer to this problem is that there are no such functions.
    Hi, there is a typo in my question. I have made the necessary changes. Sorry for that. Is my solution still correct?
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  4. #4
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    Re: Find all entire functions f

    Yes - that looks absolutely correct to me. And because your reasoning works backwards from assuming such an entire f, there are no other solutions. Well done.
    Last edited by johnsomeone; October 17th 2012 at 09:12 AM.
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