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Math Help - Residue Theorem and Real integrals

  1. #1
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    Residue Theorem and Real integrals

    Hello,

    I need help in solving the following real integral using the residue theorem.
    \int_0^\infty \frac{sin^2 x}{x^2}dx.
    What starting complex function should I use? and what contour?

    Thank you
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    One way:

    (a) Consider \int_C\dfrac{e^{iz}dz}{z},\quad C=\gamma_1\cup\gamma_2\cup \gamma_3\cup\gamma_4

    \begin{Bmatrix}\gamma_1(t)=t,\;t\in[ \epsilon,R]\\\gamma_2(t)=Re^{it},\;t\in [0,\pi]\\\gamma_3(t)=t,\;t\in [-R,- \epsilon]\\ \gamma_4(t)= \epsilon e^{(\pi-t)i},\;t\in [0,\pi]\end{matrix}

    Using the residue theorem you'll obtain \int_0^{+\infty}\dfrac{\sin x \;dx}{x}=\dfrac{\pi}{2}

    (b) Using the integration by parts method, you'll obtain

    \int_0^{+\infty}\dfrac{\sin^2 x \;dx}{x^2}=\int_0^{+\infty}\dfrac{\sin 2x \;dx}{x}

    (c) Using the substitution t=2x you'll obtain

    \int_0^{+\infty}\dfrac{\sin 2x \;dx}{x}=\int_0^{+\infty}\dfrac{\sin t \;dt}{t}=\dfrac{\pi}{2}
    Last edited by FernandoRevilla; May 17th 2011 at 09:56 AM.
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