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)
$$\begin{eqnarray}
\lim_{\Delta z\rightarrow0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}
&=&
\lim_{\Delta x\rightarrow0}\frac{\left\{u\left(x_0+\Delta x, y_0\right)+iv\left(x_0+\Delta x, y_0\right)\right\}
-\left\{u\left(x_0, y_0\right)+iv\left(x_0, y_0\right)\right\}}{\Delta x}
\;\ldots\;z_0,\Delta zin\mathbb{C},\;x_0,y_0,\Delta x\in\mathbb{R}
\\&=&
\lim_{\Delta x\rightarrow0}\left\{
\frac{
u\left(x_0+\Delta x, y_0\right)
-u\left(x_0, y_0\right)
}{\Delta x}
+i\frac{
v\left(x_0+\Delta x, y_0\right)
-v\left(x_0, y_0\right)
}{\Delta x}
\right\}
\\&=&\frac{\partial u\left(x_0, y_0\right)}{\partial x}+i\frac{\partial v\left(x_0, y_0\right)}{\partial x}
\end{eqnarray}$$
複素平面の虚軸方向の微分(偏微分)
$$\begin{eqnarray}
\lim_{\Delta z\rightarrow0}\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}
&=&
\lim_{\Delta y\rightarrow0}\frac{\left\{u\left(x_0, y_0+\Delta y\right)+iv\left(x_0, y_0+\Delta y\right)\right\}
-\left\{u\left(x_0, y_0\right)+iv\left(x_0, y_0\right)\right\}}{i\Delta y}
\;\ldots\;z_0,\Delta zin\mathbb{C},\;x_0,y_0,\Delta y\in\mathbb{R}
\\&=&
\lim_{\Delta y\rightarrow0}\left\{
\frac{
u\left(x_0, y_0+\Delta y\right)
-u\left(x_0, y_0\right)
}{i\Delta y}
+i\frac{
v\left(x_0, y_0+\Delta y\right)
-v\left(x_0, y_0\right)
}{i\Delta y}
\right\}
\\&=&\frac{1}{i}\frac{\partial u\left(x_0, y_0\right)}{\partial y}+\frac{i}{i}\frac{\partial v\left(x_0, y_0\right)}{\partial y}
\\&=&\frac{i}{i}\frac{1}{i}\frac{\partial u\left(x_0, y_0\right)}{\partial y}+\frac{\partial v\left(x_0, y_0\right)}{\partial y}
\\&=&\frac{i}{-1}\frac{\partial u\left(x_0, y_0\right)}{\partial y}+\frac{\partial v\left(x_0, y_0\right)}{\partial y}
\\&=&-i\frac{\partial u\left(x_0, y_0\right)}{\partial y}+\frac{\partial v\left(x_0, y_0\right)}{\partial y}
\\&=&\frac{\partial v\left(x_0, y_0\right)}{\partial y}+i\left\{-\frac{\partial u\left(x_0, y_0\right)}{\partial y}\right\}
\end{eqnarray}$$
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
$$\left\{
\begin{eqnarray}
\frac{\partial u}{\partial x}&=&\frac{\partial v}{\partial y}
\\\frac{\partial v}{\partial x}&=&-\frac{\partial u}{\partial y}
\end{eqnarray}
\right.$$
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