${\displaystyle {\boldsymbol {p))=\lim _{\delta \to 0}{\big \{}q\,{\boldsymbol {r))_{+}-q\,{\boldsymbol {r))_{-}{\big \))=\lim _{\delta \to 0}q\,{\boldsymbol {\delta ))}$

${\displaystyle {\begin{aligned}\rho ({\boldsymbol {x)))&=\lim _{\delta \to 0}{\Big \{}q\,\delta ^{3}({\boldsymbol {x))-{\boldsymbol {r))_{+})-q\,\delta ^{3}({\boldsymbol {x))-{\boldsymbol {r))_{-}){\Big \))\\&=-{\boldsymbol {p))\cdot \operatorname {grad} \delta ^{3}({\boldsymbol {x))-{\boldsymbol {r)))\\\end{aligned))}$

${\displaystyle \rho ({\boldsymbol {x)))=-\sum _{i}{\boldsymbol {p))_{i}\cdot \operatorname {grad} \delta ^{3}({\boldsymbol {x))-{\boldsymbol {r))_{i})}$

${\displaystyle F({\boldsymbol {x)))=\int \rho ({\boldsymbol {y)))\,K({\boldsymbol {x))-{\boldsymbol {y)))\,d^{3}y}$

${\displaystyle {\begin{aligned}F({\boldsymbol {x)))&=-\int K({\boldsymbol {x))-{\boldsymbol {y)))\,{\boldsymbol {p))\cdot \operatorname {grad} \delta ^{3}({\boldsymbol {y))-{\boldsymbol {r)))\,d^{3}y\\&=-\int {\boldsymbol {p))\cdot \operatorname {grad} K({\boldsymbol {x))-{\boldsymbol {y)))\,\delta ^{3}({\boldsymbol {y))-{\boldsymbol {r)))\,d^{3}y-\oint ({\boldsymbol {p))\cdot {\boldsymbol {n)))\,K({\boldsymbol {x))-{\boldsymbol {y)))\,\delta ^{3}({\boldsymbol {y))-{\boldsymbol {r)))\,dS\\&=-{\boldsymbol {p))\cdot \operatorname {grad} K({\boldsymbol {x))-{\boldsymbol {r)))+{\text{(boundary term)))\\\end{aligned))}$

${\displaystyle F({\boldsymbol {x)))=-{\boldsymbol {p))\cdot \operatorname {grad} K({\boldsymbol {x))-{\boldsymbol {r)))}$

${\displaystyle \phi ({\boldsymbol {x)))=-{\frac {1}{4\pi \epsilon _{0))}{\boldsymbol {p))\cdot \operatorname {grad} {\frac {1}{|{\boldsymbol {x))-{\boldsymbol {r))|))={\frac {1}{4\pi \epsilon _{0))}{\frac ((\boldsymbol {p))\cdot ({\boldsymbol {x))-{\boldsymbol {r)))}{|{\boldsymbol {x))-{\boldsymbol {r))|^{3))))$

${\displaystyle {\boldsymbol {p))({\boldsymbol {x)))=\int ({\boldsymbol {y))-{\boldsymbol {x)))\,\rho ({\boldsymbol {y)))\,d^{3}y}$

${\displaystyle \rho ({\boldsymbol {x)))=\sum _{i}q_{i}\,\delta ({\boldsymbol {x))-{\boldsymbol {r))_{i})}$

${\displaystyle {\boldsymbol {p))({\boldsymbol {x)))=\sum _{i}q_{i}\ ({\boldsymbol {r))_{i}-{\boldsymbol {x)))}$

${\displaystyle Q=\int \rho ({\boldsymbol {y)))\,d^{3}y=0}$

${\displaystyle {\boldsymbol {p))({\boldsymbol {x)))=\int {\boldsymbol {y))\,\rho ({\boldsymbol {y)))\,d^{3}y}$

${\displaystyle {\boldsymbol {p))({\boldsymbol {x)))=\sum _{i}{\boldsymbol {p))_{i))$

${\displaystyle {\boldsymbol {p))=\alpha {\boldsymbol {E))}$