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Math Help - Singularities of the function

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    Singularities of the function

    How would you find and classify all the singularities of the function f(z) = cot (z)/z^4?

    Clearly the function is undefined at 0, so has a singularity at 0, but there surely must be more to this?
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  2. #2
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    Now you have to find the order of the pole, if it's indeed one (it seems to be)

    that is to find n such that z^nf(z) has a limit as z goes to 0.
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     \lim_{z\to 0} z\cot(z) = \lim_{z\to 0 } \frac{z}{\tan(z)} = what ?
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    = 0 ?

    so the singularity at z = 0 is a simple pole? is that correct to say?
    I'm uncertain as to why it is zcot(z) and not just cot(z)/z^4?
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  5. #5
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    Quote Originally Posted by Roxanne123456789 View Post
    = 0 ?

    so the singularity at z = 0 is a simple pole? is that correct to say?
    I'm uncertain as to why it is zcot(z) and not just cot(z)/z^4?
    No. The previous poster is giving you a hint as to what the smallest integer value of n is such that \lim_{z \to 0} \left( z^n \frac{\cot z}{z^4}\right) exists and is finite.
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  6. #6
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    Also, your problem said to "find and classify all the singularities". Since cot(z)= \frac{cos(z)}{sin(z)} this function has a singularity wherever sin(z) is 0.
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