C-R of Laurent Series

In summary, there are two main methods for doing complex contour integration: using the Cauchy-Riemann formula or the Laurent series and finding the first principle coefficient. The selection criteria for choosing between these methods would typically involve whether you are summing residues or integrating around the contour, and in most cases, the residue method would be preferred.
  • #1
hob
6
0
When doing complex contour integration one can use the C-R formula or the Laurent series and find the first coefficient of the principle part. What are the selection criteria for choosing these methods?

Regards,

Hob
 
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  • #2
What do you mean by "use the Cauchy-Riemann equations to do a contour integral"? Are you referring to actually integrating around the contour as opposed to summing the residues inside the contour?
 
  • #3
Hi, I mean summing within the contour.

[tex]2 \pi[/tex] x [tex]\sum residues[/tex] = Integral around the contour.

You can also use the Laurent Expansion and finding the first principle coefficient.

I am unsure what method you would use when presented with a complex integration,

Regards,
 
Last edited:
  • #4
??The "first principle coefficient" of the Laurent expansion, around pole [itex]z_0[/itex], by which I think you mean the coefficient of [itex]z^{-1}[/itex], is the residue at that [itex]z_0[/itex]. They are the same method.
 
  • #5
Correct me if I'm wrong, but I think you mean should you try to evaluate the integral directly, or do the residue by other means? Almost certainly the residue method. After all, if you could muscle your way through integrals, you wouldn't need contour integration.
 

1. What is a Laurent series?

A Laurent series is a type of mathematical series used to represent complex functions, specifically those that are analytic (smooth and continuous). It is an infinite sum of terms that includes both positive and negative powers of a variable, such as z.

2. How is a Laurent series different from a Taylor series?

A Taylor series only includes positive powers of the variable, while a Laurent series includes both positive and negative powers. Additionally, a Taylor series is centered around a specific point, while a Laurent series can be centered around any point in the complex plane.

3. What is the region of convergence (ROC) for a Laurent series?

The ROC is the set of points in the complex plane for which the Laurent series converges. It is a ring-shaped region between two circles, known as the inner and outer radii, centered around the center of the series.

4. How is the coefficient of a Laurent series calculated?

The coefficient of each term in a Laurent series can be calculated using the formula: cn = (1/2πi) ∮ f(z)(z-z0)n-1 dz, where f(z) is the function being represented, z0 is the center of the series, and n is the power of the term.

5. What are some real-world applications of Laurent series?

Laurent series have many applications in physics, engineering, and other fields. They are used to model physical phenomena such as electromagnetic fields and fluid flow, as well as in signal processing and image reconstruction. They are also important in complex analysis and the study of singularities in functions.

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