x̄ -> Cauchy equation for Laurent series

 We talked about cauchy and laurent in the post about Normality of a family (https://kapitals-pi.blogspot.com/2021/02/normality-of-family-repost.html). 

Below are there proofs explained.

Given a Laurent series representation of a function f(z) in the annulus rj<|z−a|<rj+1, where rj and rj+1 are the radii of two consecutive circles in a Laurent arrangement centered at a, the Cauchy formula for 

Laurent series states: ck=12πi∮Cf(z)(z−a)k+1dz where: - ck is the k-th coefficient in the Laurent series expansion of f(z), - C is a positively oriented contour lying in the annulus rj<|z−a|<rj+1, - k is any integer. 

The contour C can be any simple closed curve that winds once counterclockwise around the singularity a and lies entirely within the annulus rj<|z−a|<rj+1. 

This formula essentially tells us that the coefficient ck of (z−a)k in the Laurent series expansion of f(z) is given by a contour integral around the singularity a of f(z) divided by 2πi. 

This formula is extremely useful in computing coefficients of Laurent series, especially in cases where finding the coefficients directly from the series expansion is difficult.

 Riemann’s theorem typically refers to the Riemann mapping theorem, which states that any simply connected open subset of the complex plane that is not the whole plane can be conformally mapped onto the open unit disk. 

This theorem is not directly related to removable singularities. 

However, if you’re referring to a theorem specifically about removable singularities, then it’s likely you’re talking about a result from complex analysis. 

One of the fundamental theorems regarding removable singularities is: 

Theorem (Removable Singularity Theorem): Suppose f(z) is holomorphic (analytic) on a punctured neighborhood of z=a (i.e., on 0<|z−a|<r) except possibly at z=a, where it has a singularity. 

If f(z) is bounded in a neighborhood of z=a, then the singularity at z=a is removable. 

To prove that a function f(z) has a removable singularity at z=0, one typically demonstrates that the function is holomorphic in some punctured neighborhood of z=0 and is bounded in a neighborhood of z=0.

This work is licensed under a Creative Commons Attribution 4.0 International License.