Principal Part of a Laurent Series

In summary, if f is a meromorphic function with a finite number of singularities, the principal part of the Laurent series centered at any singularity will have an infinite convergence radius. This is because the principal part can be represented as a power series in (z-z_j)^-1, which has an infinite radius of convergence. Therefore, the principal part of the Laurent series will converge for all z, and the proof is complete.
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Homework Statement


It f is a meromorphic function with finite number of singularities, prove that the the principal part of the laurent series centered at a singularity has infinite convergence radius.


Homework Equations


f(z)=Ʃ(a_n)(z-z_j) where z_j is the singularity.
Principal part = Ʃ(a_n)(z-z_j) where the sum goes from -1 to -infinity


The Attempt at a Solution


I see that the principal part is a power series in (z-z_j)^-1 but I'm not sure what else I'm supposed to be looking for.
 
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  • #2
I know that power series have infinite radius of convergence but I'm not sure how to tie that in with the problem.
 

FAQ: Principal Part of a Laurent Series

1. What is the principal part of a Laurent series?

The principal part of a Laurent series is the part of the series that contains the terms with negative powers of the variable. It is denoted by P(z) and is used to describe the behavior of a function at a singular point.

2. How is the principal part of a Laurent series calculated?

The principal part of a Laurent series can be calculated by finding the coefficients of the terms with negative powers of the variable. These coefficients can be obtained by using the formula c-n = 1/2πi ∫C f(z)(z-z0)n+1 dz, where C is a circle centered at the singular point z0.

3. What is the significance of the principal part of a Laurent series?

The principal part of a Laurent series is important because it helps us understand the behavior of a function at a singular point. It can also be used to determine the type of singularity at that point, such as a pole or an essential singularity.

4. Can the principal part of a Laurent series be infinite?

Yes, the principal part of a Laurent series can be infinite. This happens when the function has an essential singularity at the point of expansion, and the series contains infinitely many terms with negative powers of the variable.

5. How is the principal part of a Laurent series different from the Taylor series?

The principal part of a Laurent series and the Taylor series are different in that the Taylor series only contains terms with non-negative powers of the variable, while the Laurent series contains both positive and negative powers. Additionally, the Taylor series is used to approximate a function near a regular point, while the Laurent series is used to describe the behavior of a function at a singular point.

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