Contour integration using residue theorem (quick question)

Thread starter markov4
Start date Apr 28, 2010
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Integration Residue Theorem

In summary, contour integration is a technique used in complex analysis to evaluate complex integrals by integrating along a closed curve in the complex plane. The residue theorem is a powerful tool that allows for the evaluation of certain complex integrals using the residues of a function. The residue of a function can be calculated by finding the coefficient of the term with a negative power in its Laurent series expansion at a singular point inside the contour. Applications of contour integration using the residue theorem include solving problems in physics, engineering, and mathematics. However, there are limitations to using the residue theorem, such as the requirement for the integral to be expressed as a sum of simple poles inside the contour and for the function to be analytic inside the contour. The contour must also be

Apr 28, 2010

markov4

Homework Statement

The Attempt at a Solution

So i have poles at: z=-1 of order 3, z=1 and z=2. For part i), no poles are located inside the contour, therefore the residue is 0. <--is that right to say, that since there are no poles inside the contour, the residue is zero?

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Apr 28, 2010

HallsofIvy

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Homework Helper

Yes, exactly!

1. What is contour integration?

Contour integration is a technique used in complex analysis to evaluate complex integrals. It involves integrating along a closed curve in the complex plane, also known as a contour.

2. What is the residue theorem?

The residue theorem is a powerful tool in complex analysis that allows for the evaluation of certain complex integrals using the residues of a function. It states that the value of a contour integral around a closed curve is equal to the sum of the residues of the function inside the contour.

3. How is the residue of a function calculated?

The residue of a function can be calculated by finding the coefficient of the term with a negative power in the Laurent series expansion of the function at a singular point inside the contour.

4. What are some applications of contour integration using the residue theorem?

Contour integration using the residue theorem has many applications in physics, engineering, and mathematics. It is commonly used to solve problems involving complex integrals, such as calculating areas, volumes, and probabilities.

5. Are there any limitations to using the residue theorem for contour integration?

Yes, the residue theorem can only be applied to integrals that can be expressed as a sum of simple poles inside the contour. It also requires the function to be analytic, except for isolated singularities, inside the contour. Additionally, the contour must be closed and simple, meaning it does not intersect itself.