Singularity

• Jul 13th 2009, 06:50 AM
Richmond
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

$\displaystyle f(z) = \sin(\frac{1}{z})$
• Jul 13th 2009, 10:14 AM
pomp
Quote:

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;

$\displaystyle f(z)$ 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 $\displaystyle n<0$

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