تابع گرین

${\displaystyle [G(x,s){]}_{x=s^{-}}^{x=s^{+}}=0}$

${\displaystyle [\frac{\mathrm{d}}{\mathrm{d}\mathrm{x}}G(x,s){]}_{x=s^{-}}^{x=s^{+}}=\frac{1}{\mathrm{p}(\mathrm{s})}}$

${\displaystyle {\mathrm{\nabla }}^{2}G(x,s)=\delta (x-s)}$

${\displaystyle aG(x,s)+bn.{\mathrm{\nabla }}_{x}G(x,s)=0}$

${\displaystyle x=s+r}$

${\displaystyle S_{\beta }=}$

${\displaystyle x:|x-s|=r\le \beta }$

${\displaystyle {\mathrm{\nabla }}^{2}G(x,s)=\delta (x,s)}$

${\displaystyle {\int }_{S_{\beta }}{\mathrm{\nabla }}^{2}G(x,s)dx={\int }_{S_{\beta }}\delta (x,s)dx}$

${\displaystyle {\int }_{\mathrm{\partial }{S}_{\beta }}\delta (x,s).ndl=1}$

${\displaystyle \underset{ϵ\to 0}{lim}{\int }_0^{2\pi }\frac{\mathrm{\partial }\mathrm{G}}{\mathrm{\partial }\mathrm{r}}r{|}_{r=ϵ}d\theta =1}$

${\displaystyle G(x,s)=\frac{1}{2\pi }ln|x-s|}$

${\displaystyle \underset{ϵ\to 0}{lim}{\int }_0^{\pi }{\int }_0^{2\pi }\frac{\mathrm{\partial }\mathrm{G}}{\mathrm{\partial }\mathrm{r}}{r}^2{|}_{r=ϵ}sin\phi d\theta d\phi =1}$

${\displaystyle G(x,s)=\frac{-1}{4\pi |\mathrm{x}-\mathrm{s}|}}$

${\displaystyle c_L(s)u_L(s)=c_R(s)u_R(s)}$

${\displaystyle c_R(s)\frac{\mathrm{d}{\mathrm{u}}_{\mathrm{R}}}{\mathrm{d}\mathrm{x}}(s)-c_L(s)\frac{\mathrm{d}{\mathrm{u}}_{\mathrm{L}}}{\mathrm{d}\mathrm{x}}(s)=\frac{1}{\mathrm{p}(\mathrm{s})}}$

${\displaystyle c_L(s)=\frac{{\mathrm{u}}_{\mathrm{R}}(\mathrm{s})}{\mathrm{p}(\mathrm{s})\mathrm{W}(\mathrm{s})}}$

${\displaystyle \frac{{\mathrm{d}}^2\mathrm{u}(\mathrm{x})}{\mathrm{d}{x}^2}=f(x)}$

${\displaystyle {\int }_0^{L}f(s)G(x,s)ds}$

${\displaystyle \frac{{\mathrm{d}}^2\mathrm{G}(\mathrm{x},\mathrm{s})}{\mathrm{d}{x}^2}=\delta (x-s)}$

${\displaystyle bs=d(s-L)}$