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    Residues and Closed Contours : Solve the Integral

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    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.

    $2.49

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