azの微分

$az$の微分

$u+iv$で表す

$$\begin{array}{rcl}az& =& a(x+iy)\dots a,x,y\in \mathbb{R},z\in \mathbb{C}\\ & =& ax+iay\\ & =& u(x,y)+iv(x,y)\end{array}$$ $$\{\begin{array}{rcl}u(x,y)& =& ax\\ v(x,y)& =& ay\end{array}$$

$u,v$を$x,y$で偏微分する

$$\begin{array}{rcl}\frac{\partial u(x,y)}{\partial x}& =& \frac{\partial }{\partial x}ax\\ & =& a\\ \frac{\partial u(x,y)}{\partial y}& =& \frac{\partial }{\partial y}ax\\ & =& 0\\ \frac{\partial v(x,y)}{\partial x}& =& \frac{\partial }{\partial x}ay\\ & =& 0\\ \frac{\partial v(x,y)}{\partial y}& =& \frac{\partial }{\partial y}ay\\ & =& a\end{array}$$

コーシー・リーマンの関係式を満たす

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

実軸方向での微分

$$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}z}az& =& \frac{\partial u(x,y)}{\partial x}+i\frac{\partial v(x,y)}{\partial x}\\ & =& a+i0\\ & =& a\end{array}$$

虚軸方向での微分

$$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}z}az& =& \frac{\partial v(x,y)}{\partial y}-i\frac{\partial u(x,y)}{\partial y}\\ & =& a-i0\\ & =& a\end{array}$$