Does the Laurent series converge?
The Laurent series converges on the open annulus A ≡ {z : r < |z − c| < R} . Outside the annulus, the Laurent series diverges. That is, at each point of the exterior of A, the positive degree power series or the negative degree power series diverges.
Is Laurent expansion unique?
expansion of a function f(z) in an annulus r<|z−z0|. i.e. aν=bν a ν = b ν , for any integer ν , whence both expansions are identical. …
Are Laurent series 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 principal part of a Laurent expansion?
The portion of the series with negative powers of z – z 0 is called the principal part of the expansion. It is important to realize that if a function has several ingularities at different distances from the expansion point , there will be several annular regions, each with its own Laurent expansion about .
Why is a Laurent series required?
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.
What is the basic difference between Taylor and Laurent series?
A power series with non-negative power terms is called a Taylor series. In complex variable theory, it is common to work with power series with both positive and negative power terms. This type of power series is called a Laurent series.
Who discovered Laurent series?
The Laurent series expansion of an analytic function, was established by Carl Friedrich Gauss in 1843, but he never got round to publishing this work. Karl Weierstrass independently discovered it during his work to rebuild the theory of complex analysis.
What is the difference between Taylor series and Laurent series?
What is the residue of Laurent series?
The residue Res(f, c) of f at c is the coefficient a−1 of (z − c)−1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
How is residue of Laurent series calculated?