I know Laurent series has to do something when function has singularities.I am trying to get idea why Laurent series in such way.. For that i read at somewhere...
"A taylor series requires the function be defined at that point. It will not work if the center point is a singularity of the function, like 0 is for 1/z or cos(z) / z.
(I could have the technical details a bit off, but it's something close to this:)
(3) $\rightarrow$ The idea behind Laurent series is that any analytic function, regardless of singularities can be decomposed in a unique way into the sum of two analytic functions, one which vanishes at 0, and the other which vanishes at infinity.
Each of these analytic functions can be taylor expanded: the first around 0, and the other around infinity.
The first function is centered at 0, so there is a neighborhood of 0 (a disc) where it is defined. It is defined anywhere within a fixed distance R.
For the second function, you get a neighborhood of infinity where it is defined. That is, you get a disc.... but this series only works outside of that disc, rather than inside. If this disc has radius r, it works only for points z > r.
When you add these two series together, you have to be conservative about your domain. Anything outside the radius R will diverge (thanks to the first series) and anything inside the radius r will diverge (thanks to the second series). It's only the sweetspot in the middle where the sum of the two series converges.
And the intersection of the interior of a large disc and the exterior of a smaller disc is an annulus.
And if you're curious about what "taylor series at infinity" means, you do a change of coordinates w = 1/z, then take the taylor series at w=0.
Can anyone explain about marked paragraph(3).bcz i think it is best motivation for Laurent series expansion.