Singularity

**Richmond** · July 13th 2009, 06:50 AM

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

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

**pomp** · July 13th 2009, 10:14 AM

Originally Posted by Richmond

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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