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Math Help - Determine multiplicity

  1. #1
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    Determine multiplicity

    Determine the multiplicity for the root z = 1 for

    f(z)=(1-cos2\pi z)^{30}

    How exactly?
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  2. #2
    Super Member girdav's Avatar
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    Re: Determine multiplicity

    Put g(z)=1-\cos(2\pi z). Then 1 has multiplicity 2: write g(z)=(z-1)^2h(z) where h is analytic and h(1)\neq 0. How can you conclude?
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  3. #3
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    Re: Determine multiplicity

    Quote Originally Posted by girdav View Post
    Put g(z)=1-\cos(2\pi z). Then 1 has multiplicity 2: write g(z)=(z-1)^2h(z) where h is analytic and h(1)\neq 0. How can you conclude?
    Not really sure, didn't really catch your drift..
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  4. #4
    Super Member girdav's Avatar
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    Re: Determine multiplicity

    Well, since f(z)=g(z)^{30}, we can write f(z)=(z-1)^{60}h(z)^{30}, and since h(1)^{30} we have the order. Hence the power 30 wasn't the main problem, you can try to do the exercise with f(z)=(1-\cos (2\pi z))^p with p a positive integer.
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