ニュートンポテンシャルの多重極展開

\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*}

\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(\boldsymbol d)=\frac{1}{\vert\boldsymbol r-\boldsymbol d\vert}\tag{1}\end{align*}

\begin{align*}\boldsymbol d&=(x,y,z)\\\boldsymbol r&=(r_x,r_y,r_z)\end{align*}

\begin{align*}f(\boldsymbol d)&=f(x,y,z)\\&=\frac{1}{\sqrt{(r_x-x)^2+(r_y-y)^2+(r_z-z)^2}}\\&=\{(r_x-x)^2+(r_y-y)^2+(r_z-z)^2\}^{-\frac{1}{2}}\tag{2}\end{align*}

\begin{align*}f(\boldsymbol d)&=f(0,0,0)\\&\ \ \ +\left[x\frac{\partial }{\partial x}+y\frac{\partial }{\partial y}+z\frac{\partial }{\partial z}\right]f(0,0,0)\\&\ \ \ +\frac{1}{2}\left[x^2\frac{\partial^2 }{\partial x^2}+y^2\frac{\partial^2}{\partial y^2}+z^2\frac{\partial^2}{\partial z^2}+2xy\frac{\partial^2}{\partial x\partial y}+2yz\frac{\partial^2}{\partial y\partial z}+2zx\frac{\partial^2}{\partial z\partial x}\right]f(0,0,0)\\&\ \ \ +\frac{1}{3!}\cdots\\&\ \ \ +\cdots\tag{1}\end{align*}

\begin{align*}\frac{\partial}{\partial x}f(0,0,0)&=\frac{1}{2}(r_x^2+r_y^2+r_z^2)^{-\frac{3}{2}}2r_x=r_x\vert\boldsymbol r\vert^{-3}\\\frac{\partial}{\partial y}f(0,0,0)&=\frac{1}{2}(r_x^2+r_y^2+r_z^2)^{-\frac{3}{2}}2r_y=r_y\vert\boldsymbol r\vert^{-3}\\\frac{\partial}{\partial z}f(0,0,0)&=\frac{1}{2}(r_x^2+r_y^2+r_z^2)^{-\frac{3}{2}}2r_z=r_z\vert\boldsymbol r\vert^{-3}\\\frac{\partial^2}{\partial x^2}f(0,0,0)&=-(r_x^2+r_y^2+r_z^2)^{-\frac{3}{2}}+3r_x^2(r_x^2+r_y^2+r_z^2)^{-\frac{5}{2}}=-\vert\boldsymbol r\vert^{-3}+3r_x^2\vert\boldsymbol r\vert^{-5}\\\frac{\partial^2}{\partial y^2}f(0,0,0)&=-(r_x^2+r_y^2+r_z^2)^{-\frac{3}{2}}+3r_y^2(r_x^2+r_y^2+r_z^2)^{-\frac{5}{2}}=-\vert\boldsymbol r\vert^{-3}+3r_y^2\vert\boldsymbol r\vert^{-5}\\\frac{\partial^2}{\partial z^2}f(0,0,0)&=-(r_x^2+r_y^2+r_z^2)^{-\frac{3}{2}}+3r_z^2(r_x^2+r_y^2+r_z^2)^{-\frac{5}{2}}=-\vert\boldsymbol r\vert^{-3}+3r_z^2\vert\boldsymbol r\vert^{-5}\\\frac{\partial^2}{\partial x\partial y}f(0,0,0)&=3r_xr_y(r_x^2+r_y^2+r_z^2)^{-\frac{5}{2}}=3r_xr_y\vert\boldsymbol r\vert^{-5}\\\frac{\partial^2}{\partial y\partial z}f(0,0,0)&=3r_yr_z(r_x^2+r_y^2+r_z^2)^{-\frac{5}{2}}=3r_yr_z\vert\boldsymbol r\vert^{-5}\\\frac{\partial^2}{\partial z\partial x}f(0,0,0)&=3r_zr_x(r_x^2+r_y^2+r_z^2)^{-\frac{5}{2}}=3r_zr_x\vert\boldsymbol r\vert^{-5}\end{align*}

\begin{align*}f(\boldsymbol d)=\frac{1}{\vert\boldsymbol r\vert}+\frac{xr_x+yr_y+zr_ z}{\vert\boldsymbol r\vert^3}+\frac{3(xr_x+yr_ y+zr_z)^2-\vert\boldsymbol r\vert^2(r_ x+r_y+r_z)^2}{2\vert \boldsymbol r\vert^5}+\cdots\tag{3}\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\tag{4}\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\tag{4}\end{align*}

\begin{align*}\boldsymbol r\cdot\boldsymbol d=\vert\boldsymbol r\vert\vert\boldsymbol d\vert\cos\gamma\tag{5}\end{align*}

\begin{align*}\frac{1}{\vert \boldsymbol r-\boldsymbol d\vert}&=\frac{1}{\vert\boldsymbol r\vert}+\frac{\vert\boldsymbol r\vert\vert\boldsymbol d\vert\cos\gamma}{\vert\boldsymbol r\vert^3}+\frac{\vert\boldsymbol r\vert^2\vert\boldsymbol d\vert^2(3\cos^2\gamma-1)}{2\vert \boldsymbol r\vert^5}+\cdots\\&=\sum_{l=0}^\infty\frac{\vert\boldsymbol d\vert^{l}}{\vert\boldsymbol r\vert^{l+1}}P_l(\cos\gamma)\tag{6}\end{align*}

\begin{align*}\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{l=0}^\infty P_l(x)t^l\tag{7}\end{align*}

\begin{align*}f(\boldsymbol d)&=\frac{1}{\vert \boldsymbol r-\boldsymbol d\vert}\\&=\frac{1}{\sqrt{(\boldsymbol r-\boldsymbol d)\cdot(\boldsymbol r-\boldsymbol d)}}\\&=\frac{1}{\sqrt{\vert\boldsymbol r\vert^2+\vert\boldsymbol d\vert^2-2\boldsymbol r\cdot\boldsymbol d}}\\&=\frac{1}{\sqrt{\vert\boldsymbol r\vert^2+\vert\boldsymbol d\vert^2-2\vert\boldsymbol r\vert\vert\boldsymbol d\vert\cos\gamma}}\\&=\frac{1}{\vert\boldsymbol r\vert\sqrt{1+\frac{\vert\boldsymbol d\vert^2}{\vert\boldsymbol r\vert^2}-2\frac{\vert\boldsymbol d\vert}{\vert\boldsymbol r\vert}\cos\gamma}}\tag{8}\end{align*}