多重極展開

\begin{align*}f(r,\theta,\varphi)&=f(x,y,z)\\&=\sum_{n_1=0}^\infty\sum_{n_2=0}^\infty\sum_{n_3=0}^\infty\frac{(x-x’)^{n_1}(y-y’)^{n_2}(z-z’)^{n_3}}{n_1!n_2!n_3!}\frac{\partial^{n_1+n_2+n_3}}{\partial x^{n_1}\partial y^{n_2}\partial z^{n_3}}f(x’,y’,z’)\end{align*}

\begin{align*}f(r,\theta,\varphi)&=f(x,y,z)\\&=\sum_{n_1=0}^\infty\sum_{n_2=0}^\infty\sum_{n_3=0}^\infty\frac{(x-x’)^{n_1}(y-y’)^{n_2}(z-z’)^{n_3}}{n_1!n_2!n_3!}\frac{\partial^{n_1+n_2+n_3}}{\partial x^{n_1}\partial y^{n_2}\partial z^{n_3}}f(x’,y’,z’)\\&=f(x’,y’,z’)\\&\ \ \ +\left[(x-x’)\frac{\partial }{\partial x}+(y-y’)\frac{\partial }{\partial y}+(z-z’)\frac{\partial }{\partial z}\right]f(x’,y’,z’)\\&\ \ \ +\frac{1}{2}\left[(x-x’)^2\frac{\partial^2 }{\partial x^2}+(y-y’)^2\frac{\partial^2}{\partial y^2}+(z-z’)^2\frac{\partial^2}{\partial z^2}+2(x-x’)(y-y’)\frac{\partial^2}{\partial x\partial y}+2(y-y’)(z-z’)\frac{\partial^2}{\partial y\partial z}+2(z-z’)(x-x’)\frac{\partial^2}{\partial z\partial x}\right]f(x’,y’,z’)\\&\ \ \ +\frac{1}{3!}\cdots\\&\ \ \ +\cdots\tag{1}\end{align*}

\begin{align*}\frac{1}{\vert \boldsymbol r-\boldsymbol d\vert}\end{align*}

\begin{align*}\frac{1}{\vert \boldsymbol r-\boldsymbol d\vert}=\frac{1}{\vert\boldsymbol r\vert}+\frac{\vert\boldsymbol r\cdot\boldsymbol d\vert}{\vert\boldsymbol r\vert^3}+\frac{3\vert\boldsymbol r\cdot\boldsymbol d\vert^2-\vert\boldsymbol r\vert^2\vert\boldsymbol d\vert^2}{2\vert \boldsymbol r\vert^5}+\cdots\end{align*}

\begin{align*}\frac{1}{\vert \boldsymbol r-\boldsymbol d\vert}&=\sum_{l=0}^\infty\frac{\vert\boldsymbol d\vert^{l}}{\vert\boldsymbol r\vert^{l+1}}P_l(\cos\gamma)\tag{6}\end{align*}