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Math Help - a tough integral

  1. #1
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    a tough integral

    Hey guys.

    I have this problem:

    http://img12.imageshack.us/img12/6729/91881544.jpg

    It's like a two parts problem.
    The first one was to prove the equation in red, I did that with a bit of trigo.
    The second part is to use the integral of the function in green to prove the thing in yellow. I'm stuck on the path that goes from - epsilon to epsilon, that semi circle, how can I find the value of this function on the path?

    Thanks.
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  2. #2
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    Opalg's Avatar
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    When x is small, sin(x)/x is close to 1. So you are integrating a function that is approximately 1 over a very short interval (of length πε), and the integral will be small, tending to 0 as ε→0. Another way to see this is to notice that the singularity at 0 is removable, so what happens near it does not really affect the integral.
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