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Math Help - Need help calculating a Residue

  1. #1
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    Need help calculating a Residue

    I am trying to integrate log(x)/( (1+x^2)^2) and I'm stuck at trying to find the residue of that function at i. I've tried expanding it to find the coefficient of (x-i) but I just get into a mess and as the pole is a double covert pole I don't think there is any nice formula that would lead me to the answer. Can anyone help please?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kevinlightman View Post
    I am trying to integrate log(x)/( (1+x^2)^2) and I'm stuck at trying to find the residue of that function at i. I've tried expanding it to find the coefficient of (x-i) but I just get into a mess and as the pole is a double covert pole I don't think there is any nice formula that would lead me to the answer. Can anyone help please?
    Let \displaystyle f(z)=\frac{\log(z)}{(1+z^2)^2}. Then, you noted that f has pole of order 2 at i so that


    \displaystyle \begin{aligned}\underset{z=i}{\text{Res }}f(z) &= \lim_{z\to i}\frac{d}{dz}\left((z-i)^2f(z)\right)\\ &= \lim_{z\to i}\frac{z-2z\log(z)+i}{z(z+i)^3}\\ &= \frac{\pi+2i}{8}\end{aligned}
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by kevinlightman View Post
    I am trying to integrate log(x)/( (1+x^2)^2)
    Perhaps you meant:

    I=\displaystyle\int_0^{+\infty}\dfrac{\log x\;dx}{(1+x^2)^2}

    Use Drexel28's post to prove:

    I=\ldots=\dfrac{\pi\log 2}{4}


    Fernando Revilla
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by FernandoRevilla View Post
    Perhaps you meant:

    I=\displaystyle\int_0^{+\infty}\dfrac{\log x\;dx}{(1+x^2)^2}

    Use Drexel28's post to prove:
    I=\ldots=\dfrac{\pi\log 2}{4}


    Fernando Revilla
    Indeed, one can use the semi-circle contour \Gamma_R=\left\{z:|z|=R\right\}\cap \text{URHP} and use the Residue THeorem.
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