Contour Integrals, which contour to use?

• xago
In summary, a contour integral is a type of line integral used in complex analysis to evaluate the integral of a complex-valued function along a given curve, or contour. The purpose of using a contour in a contour integral is to choose a path that simplifies the integral and allows for easy calculation. The choice of contour depends on the specific function being integrated and the desired outcome, and some common techniques for choosing a contour include using the Cauchy-Goursat theorem, finding singularities of the function, and looking for symmetry in the function. Some commonly used contours in contour integrals include circles, rectangles, and keyhole contours, while other examples include semicircles, line segments, and combinations of these basic shapes. However, not
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Homework Statement

http://img404.imageshack.us/img404/3952/contf.png

The Attempt at a Solution

Is there a set of rules or postulate that refer to which contour to use for specific integrals?
I tried to use the residue theorem for the first integral but I didn't get the right answer

Last edited by a moderator:
No, there is no general rule, but it depends entirely on the integral which path to chose.
What have you done exactly for a)? Which path did you chose and what result did you get?

1. What is a contour integral?

A contour integral is a type of line integral used in complex analysis to evaluate the integral of a complex-valued function along a given curve, or contour.

2. What is the purpose of using a contour in a contour integral?

The choice of contour in a contour integral is crucial as it determines the path along which the integral is evaluated. The goal is to choose a contour that simplifies the integral and allows for easy calculation.

3. How do I choose which contour to use in a contour integral?

The choice of contour depends on the specific function being integrated and the desired outcome. Some common techniques for choosing a contour include using the Cauchy-Goursat theorem, finding singularities of the function, and looking for symmetry in the function.

4. What are some common contours used in contour integrals?

Some commonly used contours in contour integrals include circles, rectangles, and keyhole contours. Other examples include semicircles, line segments, and combinations of these basic shapes.

5. Can I use any contour for a contour integral?

No, not all contours are suitable for contour integrals. The chosen contour must be a continuous, piecewise smooth curve that does not intersect itself. It should also enclose the region of integration and avoid any singularities of the function.

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