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Residues and Closed Contours : Solve the Integral

Calculate the following integral...

Please see attached for full question.

Solution. Consider a close contour C shown above, where C consists of and a line segment from -R and R. Consider positive orientation, namely, clockwise. Choose r large enough so that are in the region covered by C.

Let . By residual Theorem, we have

...............................(1)

Note:

(1) are the only poles of .
(2)

(2) is very useful when we compute the following residues.

.....................................(2)

.....................................(3)

Now we evaluate . We know that

Note: We use an inequality

So,

When R goes to infinity, we know that

....................(4)

By (1), (2) , (3) and (4), we take limit as R goes to infinity, we can obtain

ie.,

i.e.,

i.e.,

Comparing the real part of both sides, we get

Note:

So, we get

So, by the fact of (since is an even function)

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Solution. Consider a close contour C shown above, where C consists of and a line segment from -R and R. Consider positive orientation, namely, clockwise. Choose r large enough so that are in the region covered by C.

Let . By residual Theorem, we have

...............................(1)

Note:

(1) are ...

Solution Summary

An integral of a closed contour is solved using residues are examined.

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