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Math Help - Singularity

  1. #1
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    Singularity

    Could someone explain to me the differences between essential and removable singularity?

    Like how do I decide whether the function is essential or removable singularity? for example

    f(z) = \sin(\frac{1}{z})
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  2. #2
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    Quote Originally Posted by Richmond View Post
    Could someone explain to me the differences between essential and removable singularity?
    Like how do I decide whether the function is essential or removable singularity?
    Have you been given definitions of what it means for a function to have certain singularities? If so, what is it about the definitions you don't understand?

    I can give you definitions of the two types of singularities you asked about, from these it should be obvious what the difference between them is.

    In terms of a Laurent series;

    f(z) has a removable singularity at a point a if in it's Laurent series,

    f(z) = \sum_{n=-\infty}^{\infty} b_n(z-a)^n,

    we have that b_n = 0     \forall  n<0



    f(z) has an essential singularity at a if

    b_n \neq 0 for infinitely many n<0


    Now, about 0,

     \sin(z) = \sum_{n=0}^\infty \frac{(-1)^nz^{2n+1}}{(2n+1)!}

    Substitute z for 1/z and then look at coefficients of negative powers of z, you should then see what type of singularity occurs at z=0
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