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1) Why do I need these laurent series? As I understood from Calculus 1, the taylor series around ##x_0## will always approximate a function ##f(x)## gradually better as the order ##n## increases. First it takes into consideration ##f(x_0)##, then the first order differentiated, then second and so on until it is able to perfectly predict how the function will behave over all the ##x## space around ##x_0##.

However, now in the complex plane, I see this stuff about "singularities" (points where ##f(z)## is unanalytic)... I don't understand. Doesn't this mean that the taylor series of ##f(z)## still apply everywhere EXCEPT the singularities?

2) What exactly is the "radius of convergence" for a taylor series? I thought a taylor series around ##z_0## would always converge to its function ##f(z)## regardless of what the value ##z_0## or ##z## is, provided ##f(z)## is analytic on those points.

3) What do the mathematicians mean when they use those annuluses to explain laurent series? I realize that they mean a function ##f(z)## is analytic for any ##z## inside an annulus defined by the two circles ##C_1## and ##C_2##, but then I understand little more...

I need to understand step 2) before I can understand step 3), I guess.

All help is highly appreciated!