\(wz\)の微分
\(u+iv\)で表す
$$\begin{eqnarray}
wz&=&(a+ib)(x+iy)\;\ldots\;a,b,x,y\in\mathbb{R},\;w,z\in\mathbb{C}
\\&=&ax-by+i(ay+bx)
\\&=&u(x,y)+iv(x,y)
\end{eqnarray}$$
\(u,v\)を\(x,y\)で偏微分する
$$\begin{eqnarray}
\frac{\partial u(x,y)}{\partial x}&=&\frac{\partial }{\partial x}(ax-by)
\\&=&a
\\\frac{\partial u(x,y)}{\partial y}&=&\frac{\partial }{\partial y}(ax-by)
\\&=&-b
\\\frac{\partial v(x,y)}{\partial x}&=&\frac{\partial }{\partial x}(ay+bx)
\\&=&b
\\\frac{\partial v(x,y)}{\partial y}&=&\frac{\partial }{\partial y}(ay+bx)
\\&=&a
\end{eqnarray}$$
コーシー・リーマンの関係式を満たす
$$\href{https://shikitenkai.blogspot.com/2021/07/blog-post_19.html}{
\left\{
\begin{eqnarray}
\frac{\partial u}{\partial x}&=&\frac{\partial v}{\partial y}
\\\frac{\partial u}{\partial y}&=&-\frac{\partial v}{\partial x}
\end{eqnarray}
\right.
}$$
実軸(x)方向の微分
$$\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d} z}wz&=&\href{https://shikitenkai.blogspot.com/2021/07/blog-post_19.html}{\frac{\partial u(x,y)}{\partial x}+i\frac{\partial v(x,y)}{\partial x}}
\\&=&a+ib
\\&=&w
\end{eqnarray}$$
虚軸(y)方向の微分
$$\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d} z}wz&=&\href{https://shikitenkai.blogspot.com/2021/07/blog-post_19.html}{\frac{\partial v(x,y)}{\partial y}-i\frac{\partial u(x,y)}{\partial y}}
\\&=&a-i(-b)
\\&=&a+ib
\\&=&w
\end{eqnarray}$$
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