Complex Analysis - Contour Integration

In summary, contour integration is a technique used in complex analysis to evaluate integrals of complex-valued functions along a specific path or contour in the complex plane. It is useful for evaluating complex integrals that may not be possible to solve using traditional methods and for studying the behavior of complex functions and their singularities. The contour of integration is chosen based on the specific function being integrated and should enclose the region of interest and avoid any singularities. Contour integration can also be used for real-valued functions by considering the real part of the complex function being integrated, with the contour lying on the real axis. The Cauchy-Goursat theorem is closely related to contour integration as it provides a way to evaluate certain complex integrals by
  • #1
QuantumJG
32
0
In a lecture today we evaluated a integral:

[Tex] \oint_{\Gamma} \dfrac{3z - 2}{z^2 - z} dz [/Tex]

Where,

[Tex] \Gamma = \{ z \in \mathbb{C} | |z| + |z-1| = 3 \} [/Tex]

Our lecturer evaluated it to be 6πi

I sort of understood how he did it, but he really rushed through his steps.
 
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  • #2
So what is you specific question? You have two poles in your integrand z=0 and z=1, you use Cauchy's residue theorem to do the integral and you need to compute residues. However ONLY the poles that are inside the contour are counted.

Can you draw the contour?
 

1. What is contour integration in complex analysis?

Contour integration is a technique used in complex analysis to evaluate integrals of complex-valued functions along a specific path or contour in the complex plane.

2. Why is contour integration useful?

Contour integration allows us to evaluate complex integrals that may not be possible to solve using traditional methods. It also provides a powerful tool for studying the behavior of complex functions and their singularities.

3. How do you determine the contour of integration?

The contour of integration is chosen based on the specific function being integrated. It should enclose the region of interest and avoid any singularities of the function.

4. Can contour integration be used for real-valued functions?

Yes, contour integration can be used for real-valued functions by considering the real part of the complex function being integrated. In this case, the contour will lie entirely on the real axis.

5. What is the Cauchy-Goursat theorem and how is it related to contour integration?

The Cauchy-Goursat theorem states that if a function is analytic (differentiable) within a region enclosed by a contour, then the contour integral of the function along any closed path within that region is equal to zero. This theorem is closely related to contour integration as it provides a way to evaluate certain complex integrals by considering the properties of analytic functions.

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