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Thread: Residue (Computing Integral along branch integral)

  1. #1
    Member roshanhero's Avatar
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    Residue (Computing Integral along branch integral)

    While computing this integral \int_{-1}^{1}\frac{dx}{\sqrt{1-x^2})}, I am having trouble at finding residue at -i. My work is like this a_{-1}(-i)=\frac{1}{\sqrt{2}.2e^{i3\pi/2}}, but there is mistake in here and I haven't been able to figure it out.
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  2. #2
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    Re: Residue (Computing Integral along branch integral)

    Do you need to use complex integration? First of all, the function is even, so $\displaystyle \begin{align*} \int_{-1}^1{\frac{\mathrm{d}x}{\sqrt{1 - x^2}}} &= 2 \int_0^1{ \frac{\mathrm{d}x}{\sqrt{1 - x^2}} } \end{align*}$. Then

    $\displaystyle \begin{align*} 2\int_0^1{\frac{\mathrm{d}x}{\sqrt{1 - x^2}}} &= 2\lim_{\epsilon \to 1} \int_0^{\epsilon}{\frac{\mathrm{d}x}{\sqrt{1 - x^2}}} \\ &= 2\lim_{\epsilon \to 1} \left[ \arcsin{(x)} \right] _0^{\epsilon} \\ &= 2\left[ \lim_{\epsilon \to 1} \arcsin{( \epsilon ) } - \arcsin{(0)} \right] \\ &= 2\arcsin{(1)} \\ &= 2 \left( \frac{\pi}{2} \right) \\ &= \pi \end{align*}$
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  3. #3
    Member roshanhero's Avatar
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    Re: Residue (Computing Integral along branch integral)

    Yeah, I have to solve it through complex analysis.
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