コーシー・リーマンの関係式

コーシー・リーマンの関係式

複素平面の実軸方向の微分(偏微分)

$$\begin{array}{rcl}\underset{\mathrm{\Delta }z\to 0}{lim}\frac{f(z_0+\mathrm{\Delta }z)-f(z_0)}{\mathrm{\Delta }z}& =& \underset{\mathrm{\Delta }x\to 0}{lim}\frac{\{u(x_0+\mathrm{\Delta }x,y_0)+iv(x_0+\mathrm{\Delta }x,y_0)\}-\{u(x_0,y_0)+iv(x_0,y_0)\}}{\mathrm{\Delta }x}\dots z_0,\mathrm{\Delta }zin\mathbb{C},x_0,y_0,\mathrm{\Delta }x\in \mathbb{R}\\ & =& \underset{\mathrm{\Delta }x\to 0}{lim}\{\frac{u(x_0+\mathrm{\Delta }x,y_0)-u(x_0,y_0)}{\mathrm{\Delta }x}+i\frac{v(x_0+\mathrm{\Delta }x,y_0)-v(x_0,y_0)}{\mathrm{\Delta }x}\}\\ & =& \frac{\partial u(x_0,y_0)}{\partial x}+i\frac{\partial v(x_0,y_0)}{\partial x}\end{array}$$

複素平面の虚軸方向の微分(偏微分)

$$\begin{array}{rcl}\underset{\mathrm{\Delta }z\to 0}{lim}\frac{f(z_0+\mathrm{\Delta }z)-f(z_0)}{\mathrm{\Delta }z}& =& \underset{\mathrm{\Delta }y\to 0}{lim}\frac{\{u(x_0,y_0+\mathrm{\Delta }y)+iv(x_0,y_0+\mathrm{\Delta }y)\}-\{u(x_0,y_0)+iv(x_0,y_0)\}}{i\mathrm{\Delta }y}\dots z_0,\mathrm{\Delta }zin\mathbb{C},x_0,y_0,\mathrm{\Delta }y\in \mathbb{R}\\ & =& \underset{\mathrm{\Delta }y\to 0}{lim}\{\frac{u(x_0,y_0+\mathrm{\Delta }y)-u(x_0,y_0)}{i\mathrm{\Delta }y}+i\frac{v(x_0,y_0+\mathrm{\Delta }y)-v(x_0,y_0)}{i\mathrm{\Delta }y}\}\\ & =& \frac{1}{i}\frac{\partial u(x_0,y_0)}{\partial y}+\frac{i}{i}\frac{\partial v(x_0,y_0)}{\partial y}\\ & =& \frac{i}{i}\frac{1}{i}\frac{\partial u(x_0,y_0)}{\partial y}+\frac{\partial v(x_0,y_0)}{\partial y}\\ & =& \frac{i}{-1}\frac{\partial u(x_0,y_0)}{\partial y}+\frac{\partial v(x_0,y_0)}{\partial y}\\ & =& -i\frac{\partial u(x_0,y_0)}{\partial y}+\frac{\partial v(x_0,y_0)}{\partial y}\\ & =& \frac{\partial v(x_0,y_0)}{\partial y}+i\{-\frac{\partial u(x_0,y_0)}{\partial y}\}\end{array}$$

複素平面の各軸微分結果の実部同士，虚部同士が等しくなる場合という関係

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