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Math Help - Proving the simple poles

  1. #1
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    Proving the simple poles

    How to prove that this two functions have simple poles:

    f(z)=\frac{cos(z)}{1-2sin(z)} at z=\pi/6

    g(z)=\frac{senh(z)}{cosh(z)-1} at z=2ki\pi where k is an integer

    Regards.
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  2. #2
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    Well, what is the definition of "simple pole"?
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  3. #3
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    you have a simple pole when the maximum power of 1/z you have is 1 in the laurent series of the function
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  4. #4
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    I'm not sure I understood your definition of a 'simple pole'. Is not a pole commonly defined as the values of z such that the denominator is zero? And perhaps by simple, you mean that the pole has multiplicity one?

    For example, the first function f(z) would have a pole when 1-2\sin(z)=0, or equivalently \sin(z)=\frac{1}{2}. Then z=\frac{\pi}{6} certainly is a pole, but since \sin(z) is periodic, there most certainly are other poles also.
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