Laurent series in different regions

1. How do you find the region of convergence of Laurent series?
2. Where can I find all of Laurent series?
3. Is Laurent series expansion unique?
4. What is the advantage of Laurent series over Taylor series?

How do you find the region of convergence of Laurent series?

Its radius of convergence around z=−2, and therefore the radius of convergence of your Laurent series, is simply the distance from −2 to the nearest non-differentiable point, i.e. z=−1.

Where can I find all of Laurent series?

Determine the Laurent series for the function, f(z) = (z+1)/z around z₀ = 0. Also, define the region where the function is valid. Hence, f(z) = 1+ (1/z) is the Laurent series, which is valid on the infinite region 0 < |z| < ∞.

Is Laurent series expansion unique?

This series is unique. Proof. Fix r1,r2 with R1 < r1 < r2 < R2. Denote by γ1 and γ2 the two circles traced counterclockwise with radius r1 and r2 respectively, and note that they are homotopic in the annulus.

What is the advantage of Laurent series over Taylor series?

The method of Laurent series expansions is an important tool in complex analysis. Where a Taylor series can only be used to describe the analytic part of a function, Laurent series allows us to work around the singularities of a complex function.