sin(z)の微分

\(\sin{\left(z\right)}\)の微分

\(u+iv\)で表す

$$\begin{eqnarray} \sin{\left(z\right)} &=&\sin{\left(x\right)}\cos{\left(iy\right)}+\cos{\left(x\right)}\sin{\left(iy\right)} \\&=&\sin{\left(x\right)}\cosh{\left(y\right)}+\cos{\left(x\right)}i\sinh{\left(y\right)} \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/cosi-x-sini-x-cos-sin.html}{\cos\left(iy\right)=i\cosh\left(y\right),\;\sin\left(iy\right)=i\sinh\left(y\right)} \\&=&\sin{\left(x\right)}\cosh{\left(y\right)}+i\cos{\left(x\right)}\sinh{\left(y\right)} \\&=&u(x,y)+iv(x,y) \end{eqnarray}$$ $$\left\{\begin{eqnarray} u(x,y)&=&\sin{\left(x\right)}\cosh{\left(y\right)} \\v(x,y)&=&\cos{\left(x\right)}\sinh{\left(y\right)} \end{eqnarray}\;\ldots\;x,y\in\mathbb{R}\right.$$

\(u,v\)を\(x,y\)で偏微分する

$$\begin{eqnarray} \frac{\partial u(x,y)}{\partial x} &=&\frac{\partial}{\partial x}\sin{\left(x\right)}\cosh{\left(y\right)} \;\ldots\;x,y\in\mathbb{R} \\&=&\cosh{\left(y\right)}\frac{\partial}{\partial x}\sin{\left(x\right)} \\&=&\cosh{\left(y\right)}\cos{\left(x\right)} \;\ldots\;\frac{\mathrm{d}}{\mathrm{d}\theta}\sin{\left(\theta\right)}=\cos{\left(\theta\right)} \\&=&\cos{\left(x\right)}\cosh{\left(y\right)} \end{eqnarray}$$ $$\begin{eqnarray} \frac{\partial u(x,y)}{\partial y} &=&\frac{\partial}{\partial y}\sin{\left(x\right)}\cosh{\left(y\right)} \;\ldots\;x,y\in\mathbb{R} \\&=&\sin{\left(x\right)}\frac{\partial}{\partial y}\cosh{\left(y\right)} \\&=&\sin{\left(x\right)}\sinh{\left(y\right)} \;\ldots\;\frac{\mathrm{d}}{\mathrm{d}\theta}\cosh{\left(\theta\right)}=\sinh{\left(\theta\right)} \end{eqnarray}$$ $$\begin{eqnarray} \frac{\partial v(x,y)}{\partial x} &=&\frac{\partial}{\partial x}\cos{\left(x\right)}\sinh{\left(y\right)} \;\ldots\;x,y\in\mathbb{R} \\&=&\sinh{\left(y\right)}\frac{\partial}{\partial x}\cos{\left(x\right)} \\&=&\sinh{\left(y\right)}\left(-\sin{\left(x\right)}\right) \;\ldots\;\frac{\mathrm{d}}{\mathrm{d}\theta}\cos{\left(\theta\right)}=-\sin{\left(\theta\right)} \\&=&-\sin{\left(x\right)}\sinh{\left(y\right)} \end{eqnarray}$$ $$\begin{eqnarray} \frac{\partial v(x,y)}{\partial y} &=&\frac{\partial}{\partial y}\cos{\left(x\right)}\sinh{\left(y\right)} \;\ldots\;x,y\in\mathbb{R} \\&=&\cos{\left(x\right)}\frac{\partial}{\partial y}\sinh{\left(y\right)} \\&=&\cos{\left(x\right)}\cosh{\left(y\right)} \;\ldots\;\frac{\mathrm{d}}{\mathrm{d}\theta}\sinh{\left(\theta\right)}=\cosh{\left(\theta\right)} \end{eqnarray}$$

コーシー・リーマンの関係式を満たす

$$\href{https://shikitenkai.blogspot.com/2021/07/blog-post_19.html}{\left\{ \begin{eqnarray} \frac{\partial u}{\partial x}&=&\frac{\partial v}{\partial y} \\\frac{\partial v}{\partial x}&=&-\frac{\partial u}{\partial y} \end{eqnarray} \right.}$$

実軸方向での微分

$$\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}z}\sin{\left(z\right)} &=&\href{https://shikitenkai.blogspot.com/2021/07/blog-post_19.html}{\frac{\partial u(x,y)}{\partial x}+i\frac{\partial v(x,y)}{\partial x}} \\&=&\cos{\left(x\right)}\cosh{\left(y\right)}+i\left(-\sin{\left(x\right)}\sinh{\left(y\right)}\right) \\&=&\cos{\left(x\right)}\cosh{\left(y\right)}-i\sin{\left(x\right)}\sinh{\left(y\right)} \\&=&\cos{\left(x\right)}\cos{\left(iy\right)}-\sin{\left(x\right)}\sin{\left(iy\right)} \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/cosi-x-sini-x-cos-sin.html}{\cos\left(iy\right)=\cosh\left(y\right),\;\sin\left(iy\right)=i\sinh\left(y\right)} \\&=&\cos{\left(x+iy\right)} \\&=&\cos{\left(z\right)} \end{eqnarray}$$