∫ d 3 x 1 | x | e − i q ⋅ x = lim u → 0 ∫ d 3 x exp ( − u | x | ) | x | e − i q ⋅ x = lim u → 0 ∫ 0 ∞ d r ∫ − 1 1 d cos ϑ 2 π r 2 exp ( − u r ) r e − i q r cos ϑ = 2 π lim u → 0 ∫ 0 ∞ d r ∫ − 1 1 d s r exp ( − r ( u + i q s ) ) = 2 π − i q lim u → 0 ∫ 0 ∞ d r [ exp ( − r ( u + s i q ) ) ] − 1 1 = 2 π i q lim u → 0 [ 1 u + s i q ] − 1 1 = 2 π i q 2 i q = 4 π q 2 {\displaystyle {\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} qr\cos \vartheta }\\&=&2\pi \lim _{u\rightarrow 0}\int _{0}^{\infty }\mathrm {d} r\int _{-1}^{1}\mathrm {d} sr\exp \left(-r\left(u+\mathrm {i} qs\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}}}
![{\displaystyle {\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} qr\cos \vartheta }\\&=&2\pi \lim _{u\rightarrow 0}\int _{0}^{\infty }\mathrm {d} r\int _{-1}^{1}\mathrm {d} sr\exp \left(-r\left(u+\mathrm {i} qs\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53bcc9254ba865ad5c81f4a235b85c78e219cdf4)