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Math Help - Finding and classifying singularities

  1. #1
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    Finding and classifying singularities

    I thought I had a grasp on this topic, but then they threw in this question:

    Find and classify the singularities of
    π cot(πz )
    Z^2

    With the use of cot I'm confused which techniques to use because I know what to use if it's sin(z) or cos(z) but not cos(z)/sin(z)

    Any help insight or tricks would be greatly appreciated.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by shadowdom View Post
    I thought I had a grasp on this topic, but then they threw in this question:

    Find and classify the singularities of
    π cot(πz )
    Z^2

    With the use of cot I'm confused which techniques to use because I know what to use if it's sin(z) or cos(z) but not cos(z)/sin(z)

    Any help insight or tricks would be greatly appreciated.
    But you don't have to worry oO

    If you're not used to the cot function, you can substitute by cos/sin.
    Singularities appear at points where the function is not defined, and whether it's cot or cos/sin, they're the same.

    \frac{\pi \cot(\pi z)}{z^2}=\frac{\pi \cos(\pi z)}{\sin(\pi z) z^2}
    0 is obviously a singularity (not necessarily a pole because of sin(pi z))

    \sin(x)=0 \Leftrightarrow x=k \pi
    So here, there are also singularities for any z \in \mathbb{Z} (the set of integers)
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  3. #3
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    Cheers that does help, but how would I continue to classify each point?
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