sin(z)の微分

$\sin \left(z\right)$の微分

$u+iv$で表す

$$\begin{array}{rcl}\sin \left(z\right)& =& \sin \left(x\right)\cos \left(iy\right)+\cos \left(x\right)\sin \left(iy\right)\\ & =& \sin \left(x\right)\cosh \left(y\right)+\cos \left(x\right)i\sinh \left(y\right)\\ & & \dots \cos \left(iy\right)=i\cosh \left(y\right),\sin \left(iy\right)=i\sinh \left(y\right)\\ & =& \sin \left(x\right)\cosh \left(y\right)+i\cos \left(x\right)\sinh \left(y\right)\\ & =& u(x,y)+iv(x,y)\end{array}$$ $$\{\begin{array}{rcl}u(x,y)& =& \sin \left(x\right)\cosh \left(y\right)\\ v(x,y)& =& \cos \left(x\right)\sinh \left(y\right)\end{array}\dots x,y\in \mathbb{R}$$

$u,v$を$x,y$で偏微分する

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

コーシー・リーマンの関係式を満たす

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

実軸方向での微分

$$\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}z}\sin \left(z\right)& =& \frac{\partial u(x,y)}{\partial x}+i\frac{\partial v(x,y)}{\partial x}\\ & =& \cos \left(x\right)\cosh \left(y\right)+i(-\sin \left(x\right)\sinh \left(y\right))\\ & =& \cos \left(x\right)\cosh \left(y\right)-i\sin \left(x\right)\sinh \left(y\right)\\ & =& \cos \left(x\right)\cos \left(iy\right)-\sin \left(x\right)\sin \left(iy\right)\\ & & \dots \cos \left(iy\right)=\cosh \left(y\right),\sin \left(iy\right)=i\sinh \left(y\right)\\ & =& \cos (x+iy)\\ & =& \cos \left(z\right)\end{array}$$