User talk:Edsanville

$$\begin{array}{lll} \int \mathrm{d}^3 x \frac{1}{|\mathbf{x}|} e^{- \mathrm{i} \mathbf{q} \cdot \mathbf{x}} & = & \lim_{u \rightarrow 0} \int \mathrm{d}^3 x \frac{\exp \left( - u \left| \mathbf{x} \right| \right)}{\left| \mathbf{x} \right|} e^{- \mathrm{i} \mathbf{q} \cdot \mathbf{x}}\\ & = & \lim_{u \rightarrow 0} \int_0^\infty \mathrm{d} r \int_{-1}^1 \mathrm{d} \cos \vartheta 2 \pi r^2 \frac{\exp \left( - ur \right)}{r} e^{- \mathrm{i} q r \cos \vartheta}\\ & = & 2 \pi \lim_{u \rightarrow 0} \int_0^\infty \mathrm{d} r \int_{-1}^1 \mathrm{d}s r \exp \left( - r \left( u + \mathrm{i} q s \right) \right)\\ & = & \frac{2 \pi}{-\mathrm{i} q} \lim_{u \rightarrow 0} \int_0^\infty \mathrm{d} r \left[ \exp \left( - r \left( u + s \mathrm{i} q \right) \right)\right]_{-1}^1\\ & = & \frac{2 \pi \mathrm{i}} {q} \lim_{u \rightarrow 0} \left[ \frac{1}{u + s \mathrm{i} q } \right]_{-1}^1 \\ & = & \frac{2 \pi \mathrm{i}} {q} \frac{2} {\mathrm{i} q} \\ & = & \frac{4 \pi}{q^2 } \end{array}$$

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