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Volume integral

  1. Nov 18, 2007 #1
    1. The problem statement, all variables and given/known data

    I'm trying to understand the solution to this integral

    [itex]\int_{\mathbb{R}^3} \frac{e^{i \mathbf{x} \cdot \mathbf{a}}}{\sqrt{r^2+1}}d\mathbf{x}[/itex]

    where [itex]d\mathbf{x} =dxdydz, r = \sqrt{x^2+y^2+z^2},\mathbf{a}\in \mathbb{R}^3[/itex]

    3. The attempt at a solution

    [itex]\int_{\mathbb{R}^3} \frac{e^{i \mathbf{x} \cdot \mathbf{a}}}{\sqrt{r^2+1}}d\mathbf{x} = 2\pi \int_0^{\infty}\frac{1}{\sqrt{r^2 +1}}\frac{e^{iua}-e^{-iua}}{iua}u^2du[/itex]

    Could anyone please explain to me how this first step was obtained?
    Last edited: Nov 18, 2007
  2. jcsd
  3. Nov 18, 2007 #2


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    Just write [itex] \vec{x} \cdot \vec{a} = x a cos \theta [/itex] and the volume element as [itex] d^3 x =x^2 dx d\phi d \cos \theta [/itex] and integrate over cos theta from -1 to 1.
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