آنالیز مختلط

${\displaystyle f:\mathbb{C}\to \mathbb{C},x+iy\mapsto f(x+iy)=u(x,y)+iv(x,y).}$

${\displaystyle f^{\prime }(z_0)=\underset{z\to z_0}{lim}\frac{f(z)-f(z_0)}{z-z_0}}$

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}=\frac{\mathrm{\partial }v}{\mathrm{\partial }y},\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{\mathrm{\partial }v}{\mathrm{\partial }x}}$

${\displaystyle f(z)=\frac{1}{2\pi i}\oint \frac{f(z^{\prime })}{z^{\prime }-z}d{z}^{\prime }}$