I try to show that $$ \int\limits_{R^3} \frac{e^{i\xi x} d\xi}{\xi^2 - k^2 - i0} = e^{ikx} \int\limits_{R^3} \frac{e^{i\xi x}d\xi}{\xi^2 + 2(k + i0\frac{k}{|k|})\xi}, \;\;\; k,x \in \mathbb R^3 $$ I tried to make the change $\xi \mapsto \xi + k$ in second integral. I obtained $$ \lim\limits_{\epsilon \to 0} \; \int\limits_{\mathbb R^d} \frac{e^{i\xi x} d\xi}{(\xi + i\epsilon \frac{k}{|k|})^2 - k^2 + \epsilon^2 - 2i|k|\epsilon} $$ So I don't know what to do...
Fourier transform in $\mathbb R^3$
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fourier-analysis
integral-transforms
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0Why write out $i0$? Isn't that just zero? – 2011-11-08
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0$\xi^2-k^2 - i 0$ is a way of saying that you consider $\lim_{\epsilon \downarrow 0} \frac{e^{i\xi x} }{\xi^2 - k^2 - i \epsilon} \mathrm{d} \xi$. – 2011-11-08
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0foil the square in the denominator ;) – 2011-11-08