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Math Help - pole

  1. #1
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    pole

    Consider  f(z) = \frac{e^{z}(1- \cos(z))^{2}}{\sin (z)^{3} z^{3}}

    Now the numerator has 4 zeroes. And the denominator has 6 zeroes? And  \sin z and  z have triple zeroes in the denominator?

    Thus  f has a double pole at  0 ?

    Is this correct?
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  2. #2
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    Quote Originally Posted by manjohn12 View Post
    Consider  f(z) = \frac{e^{z}(1- \cos(z))^{2}}{\sin (z)^{3} z^{3}}

    Now the numerator has 4 zeroes. And the denominator has 6 zeroes? And  \sin z and  z have triple zeroes in the denominator?

    Thus  f has a double pole at  0 ?

    Is this correct?
    I don't know what you mean by a "zero", but it is true that 0 is a zero of the numerator with multiplicity 4, and multiplicity 6 in the denominator.
    If it is indeed a pole (you have to prove it by finding the Laurent series of this function in a neighbourhood of 0), it is likely to be a double pole...
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