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

$\displaystyle f(z) = \sin(\frac{1}{z})$

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- Jul 13th 2009, 06:50 AMRichmondSingularity
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

$\displaystyle f(z) = \sin(\frac{1}{z})$ - Jul 13th 2009, 10:14 AMpomp
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;

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

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

we have that $\displaystyle b_n = 0 $ $\displaystyle \forall n<0$

$\displaystyle f(z)$ has an essential singularity at a if

$\displaystyle b_n \neq 0$ for infinitely many $\displaystyle n<0$

Now, about 0,

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

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