Complex Integrals - Poles of Integration Outside the Curve

In summary, a complex integral involves integrating functions of a complex variable, similar to standard integration in calculus. Poles of integration outside the curve are points in the complex plane where the function being integrated is undefined and can affect the value and convergence of the integral. These poles can be handled using the residue theorem, which involves calculating the sum of the residues of the function at its poles. In some cases, poles of integration outside the curve can be avoided by choosing a specific path of integration, but in other cases, they must be considered when evaluating the integral.
  • #1
SirFibonacci
2
0

Homework Statement



[itex]\int_{|z-2i|=2}[/itex] = [itex]\frac{dz}{z^2-9}[/itex]



2. The attempt at a solution

I know that the contour described by |z-2i|=2 is a circle with a center of (0,2) (on the complex plane) with a radius of 2. The singularities of the integral fall outside of the contour (z+3 and z-3). In this case, is the solution just going to be zero or undefined?
 
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  • #2
Your integrand is analytic in a region that contains your contour, what do you know about a closed contour integral of an analytic function?
 
  • #3
The integral is 0. Makes sense. Thanks.
 

FAQ: Complex Integrals - Poles of Integration Outside the Curve

What is a complex integral?

A complex integral is a type of integration that involves functions of a complex variable. It is similar to the standard integral in calculus, but instead of integrating over real numbers, it integrates over complex numbers.

What are poles of integration outside the curve?

Poles of integration outside the curve refer to points in the complex plane where the function being integrated has a singularity or is undefined. These points lie outside the curve of integration and can affect the value of the integral.

Why do poles of integration outside the curve matter?

Poles of integration outside the curve are important because they can lead to the integral being undefined or having a different value than expected. They also affect the convergence of the integral and can change the path of integration.

How do you handle poles of integration outside the curve?

Poles of integration outside the curve can be handled by using the residue theorem, which states that the value of the integral is equal to the sum of the residues of the function at its poles. The residues can be calculated using complex analysis techniques.

Can poles of integration outside the curve be avoided?

In some cases, poles of integration outside the curve can be avoided by carefully choosing the path of integration. However, in other cases, they are unavoidable and must be taken into consideration when evaluating the integral.

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