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Math Help - complex variable

  1. #1
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    Question complex variable

    Hey, I would really appreciate some hints on how to tackle this please:

    For a natural number N, let CN be the square with vertices at (N + 1
    2 )(1 i).
    Show that there is a constant K so that
    maxzeCN | cot(pi*z)| is less than or equal to K.

    I know it could help to estimate the value of | cot(pi*z)| separately on each side of the square separately using the fact that
    cot(pi*z)= i(exp(i*pi*z) + exp(-i*pi*z))/exp(i*pi*z) - exp(-i*pi*z), but do not know how to go about this.

    Thanks.
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  2. #2
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    Quote Originally Posted by dazed View Post
    Hey, I would really appreciate some hints on how to tackle this please:

    For a natural number N, let CN be the square with vertices at (N + 1
    2 )(1 i).
    Show that there is a constant K so that
    maxzeCN | cot(pi*z)| is less than or equal to K.

    I know it could help to estimate the value of | cot(pi*z)| separately on each side of the square separately using the fact that
    cot(pi*z)= i(exp(i*pi*z) + exp(-i*pi*z))/exp(i*pi*z) - exp(-i*pi*z), but do not know how to go about this.

    Thanks.
    For |x|\leq n + \tfrac{1}{2} we have that if \alpha = x \pm i (n + \tfrac{1}{2}) is on the horizontal sides. If taken with + then it is the top and if taken with - then it is the bottom.

    This means, \left| \cot \pi \alpha \right| = \left| \frac{\cos \pi \alpha}{\sin \pi \alpha} \right| = \left| \frac{e^{\pi i \alpha} + e^{- \pi i \alpha}}{ e^{\pi i\alpha} - e^{-\pi i \alpha} } \right| \leq \frac{| e^{\pi i \alpha} | + |e^{-\pi i \alpha} | }{ ||e^{\pi i \alpha}| - |e^{-\pi i \alpha}||} = \frac{e^{\pi(n+1/2)} + e^{\pi(n-1/2)}}{e^{\pi(n+1/2)} - e^{\pi (n-1/2)}} = \coth \pi (n + \tfrac{1}{2})

    The function f(x) = \coth \pi x,x>0 is a decreasing function therefore, \coth \pi (n+\tfrac{1}{2}) \leq \coth \tfrac{3\pi}{2}.
    This means that \left| \cot \pi \alpha \right| \leq \coth \tfrac{3\pi}{2} for \alpha = x \pm i (n + \tfrac{1}{2}) where |x| \leq n+\tfrac{1}{2}

    Now if \beta = \pm (n+\tfrac{1}{2}) + i x with + being the right vertical side and - being the left vertical side then,
    |\cot \pi \beta| = | \cot \pi (n + \tfrac{1}{2} + ix) | = | \cot (\pi n + \tfrac{\pi}{2} + i\pi x) | = |\cot ( \tfrac{\pi}{2} + i\pi x )| = |-\tan (i\pi x)| = |\tanh \pi x|
    However, by property of hyperbolic tangent we have |\tanh \pi x| \leq 1.

    Therefore, \cot \pi z is bounded on C_n by the constant \max \{ 1, \coth \tfrac{3\pi}{2} \}.
    Last edited by ThePerfectHacker; January 6th 2009 at 01:58 PM.
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  3. #3
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    Thanks so much for your help, though i'm still alil confused.

    Doesn't cot(pi*z) = cos(pi*z)/sin(pi*z)?

    x
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  4. #4
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    Quote Originally Posted by dazed View Post
    Thanks so much for your help, though i'm still alil confused.

    Doesn't cot(pi*z) = cos(pi*z)/sin(pi*z)?

    x
    It does! But the proof still works out because I accidently written them the other way around.
    It is fixed now.
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  5. #5
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    Is there an i missing as well, so that:

    cot(pi*z)=i(exp(i*pi*z)+exp(-pi*i*z))/exp(pi*i*z)-exp(-pi*i*z)?

    x
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  6. #6
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    Quote Originally Posted by dazed View Post
    Is there an i missing as well, so that:

    cot(pi*z)=i(exp(i*pi*z)+exp(-pi*i*z))/exp(pi*i*z)-exp(-pi*i*z)?

    x
    The equation is put into absolute value terms and  | i | = 1.
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