wzの微分

$wz$の微分

$u+iv$で表す

$$\begin{array}{rcl}wz& =& (a+ib)(x+iy)\dots a,b,x,y\in \mathbb{R},w,z\in \mathbb{C}\\ & =& ax-by+i(ay+bx)\\ & =& u(x,y)+iv(x,y)\end{array}$$

$u,v$を$x,y$で偏微分する

$$\begin{array}{rcl}\frac{\partial u(x,y)}{\partial x}& =& \frac{\partial }{\partial x}(ax-by)\\ & =& a\\ \frac{\partial u(x,y)}{\partial y}& =& \frac{\partial }{\partial y}(ax-by)\\ & =& -b\\ \frac{\partial v(x,y)}{\partial x}& =& \frac{\partial }{\partial x}(ay+bx)\\ & =& b\\ \frac{\partial v(x,y)}{\partial y}& =& \frac{\partial }{\partial y}(ay+bx)\\ & =& a\end{array}$$

コーシー・リーマンの関係式を満たす

$$\{\begin{array}{rcl}\frac{\partial u}{\partial x}& =& \frac{\partial v}{\partial y}\\ \frac{\partial u}{\partial y}& =& -\frac{\partial v}{\partial x}\end{array}$$

実軸(x)方向の微分

$$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}z}wz& =& \frac{\partial u(x,y)}{\partial x}+i\frac{\partial v(x,y)}{\partial x}\\ & =& a+ib\\ & =& w\end{array}$$

虚軸(y)方向の微分

$$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}z}wz& =& \frac{\partial v(x,y)}{\partial y}-i\frac{\partial u(x,y)}{\partial y}\\ & =& a-i(-b)\\ & =& a+ib\\ & =& w\end{array}$$