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Math Help - Prove an entire function is linear

  1. #1
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    Prove an emtire function is linear

    Please help with this problem:

    let f be an entire function such that f(5z) = 5f(z). Prove that f is linear.

    Thanks
    Last edited by conmeo; November 30th 2009 at 09:46 PM.
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  2. #2
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    If f is entire than it has a Taylor series as it's Laurent series. Then what can be said about the expression (just write out a few terms on each side and equate coefficients and then figure what they have to be to obtain equality):

    \sum_{n=0}^{\infty} a_n(5z)^n=5\sum_{n=0}^{\infty}a_n z^n
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  3. #3
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    Thank you for your response. I still have 3 questions for this problem. Please help. Thank you.

    1. When I write the terms out, do I match term by term?
    2. the coefficient a_n on the left side is the same as the coefficient on the right side?
    3. Do I need to use the chain rule to find a_n for f(5z)?
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  4. #4
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    Quote Originally Posted by conmeo View Post
    Thank you for your response. I still have 3 questions for this problem. Please help. Thank you.

    1. When I write the terms out, do I match term by term?
    2. the coefficient a_n on the left side is the same as the coefficient on the right side?
    3. Do I need to use the chain rule to find a_n for f(5z)?
    Yes, yes, no. Just write:

    a_0+5 a_1 z+25 a_2 z^2+\cdots=5(a_0+a_1 z+a_2 z^2+\cdots) and in order for that to hold, all the a_n have to be zero except a_1. That implies the function is linear.
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  5. #5
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    since it's a Taylor series, Do I have to write the n! in the denominator?
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  6. #6
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    Quote Originally Posted by conmeo View Post
    since it's a Taylor series, Do I have to write the n! in the denominator?
    Merely note that since both the LHS and the RHS converge we may rewrite shawsend's equation as \sum_{n=1}^{\infty}z^n\left(5a_n-5^na_n\right)=0. Clearly though for the LHS to equal zero we need that 5a_n-5^na_n=0\quad \forall n. It is trivially true for n=1 for n>1 it will only be true if a_n=0. Therefore f(z)=5\sum_{n=1}^{\infty}a_nz^n=5a_1z
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