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Math Help - Evaluation of a real integral using Contour Integration

  1. #1
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    Evaluation of a real integral using Contour Integration

    I'm having difficulty in using contour Integration to evaluate the integral:

    \int_{0}^\infty \frac{cos(x)}{x^2 + a^2} dx where a>0.

    Firstly I found that the function being integrated is an even function so this is equivalent to finding:

    \frac{1}{2} \int_{-\infty}^\infty \frac{cos(x)}{x^2 + a^2} dx

    Also cos(x) = Re( exp(ix)) where Re dentoes the Real part of exp(ix) function. The integral in question is now:
    \frac{1}{2}Re\int_{-\infty}^\infty \frac{exp(ix)}{x^2 + a^2} dx
    so the complex function in question is:
    f(z)=\frac{exp(iz)}{z^2 + a^2} which is holomorphic everywhere in the complex plane except for the points ai, and -ai. That is what I've found so far.

    What contour do I have to use? Also how would you integrate it because I'm seeing there will be much difficulty integrating \frac{exp(iz)}{z^2 + a^2} with the z being replaced by the suitable parameterization.
    Thanks
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  2. #2
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    Also apologies if this thread is in the wrong section. I wasn't sure whether to put this under calculus or something else...
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  3. #3
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    Make a semicircle that includes the real axis. You can choose either one that encloses z=ia or z=-ia. Use the parametrization z=Re^{it} for the semicircle (z=ia) and show what happens when R goes to infinity.
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  4. #4
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    Does it have to be a semi cirlce? Why not a full circle?
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  5. #5
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    Quote Originally Posted by garunas View Post
    Does it have to be a semi cirlce? Why not a full circle?
    Because you want to use the residue theorem to find the value of the of the integral on

    C_1 The part on the x axis (this is what you are looking for) and

    C_2 the part on the circle above (or below) the x axis. You will want to show that this is equal to zero then

    \oint_{C_1+C_2}f(z)dz=\int_{C_1}f(z)dz+\int_{C_2}f  (z)dx

    If the 2nd integral goes to zero and the first integral is over the x-axis you have found the value you are looking for.

    See this example on wiki

    Residue theorem - Wikipedia, the free encyclopedia
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  6. #6
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    That really helped a lot! Thank you
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