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What do I have to do, or what technique do I have to use, to perform the following integral?

$ \frac{1}{4\pi^{2}}\iint_{-\infty}^{\infty}\mathrm{d}x\mathrm{d}y (x^{2}+y^{2}) \left| E(x,y) \right|^{2} = \dfrac{1}{4\pi^{2}}\iint_{-\infty}^{\infty}\mathrm{d}k_{x}\mathrm{d}k_{y} \left( \left| \frac{\partial A}{\partial x} \right|^{2} + \left| \dfrac{\partial A}{\partial y} \right|^{2} \right)$

where:

$ E(x,y) = \iint_{-\infty}^{\infty} A(k_{x},k_{y}) \exp\left[i\left(k_{x}x+k_{y}y+ z \left( k - \frac{k_{x}^{2}+k_{y}^{2}}{2k}\right)\right)\right] \mathrm{d}k_{x}\mathrm{d}k_{y} $

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    @Rodrigo: About migrating to physics.se, I think the community moderators might be able to do that; us "normal" moderators have only two options for suggesting migration, stats.se and meta.math.se -- I'm not sure why that is; perhaps someone who knows might comment? However, again, I wasn't suggesting such a migration.2011-05-29

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Assuming that what you're asking is how to get the identity in the first line, given the Fourier decomposition in the second line: Note that multiplying by $x$ in real space corresponds to applying $\mathrm i\partial/\partial k_x$ in Fourier space. You can write $x^2|E(x,y)|$ as $|xE(x,y)|^2$, and then use Parseval's theorem to replace this by the integral over the squared magnitude of the Fourier transform, where the Fourier transform is given by

$\mathrm i\frac{\partial}{\partial k_x}\left(A(k_x,k_y)\exp\left[\mathrm iz \left( k - \frac{k_{x}^{2}+k_{y}^{2}}{2k}\right)\right]\right)\;.$

Differentiating $A$ gives you the term you want in the result, and differentiating the other factor gives a factor

$\frac{\partial}{\partial k_x}\left(k - \frac{k_{x}^{2}+k_{y}^{2}}{2k}\right)=\frac{k_x}{k}-\frac{k_x}{k}+\frac{(k_x^2+k_y^2)k_x}{2k^3}=\frac{k_x}{2k}\frac{k_x^2+k_y^2}{k^2}\;.$

I guess the last term is dropped because of your approximation $k_{x}^{2}+k_{y}^{2} \ll k^{2}$?

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    Ok, now I underst$a$nd what do you do.2011-05-29