cot(z)をu(x,y)+iv(x,y)で表す

$$\begin{array}{rcl}\cot \left(z\right)& =& \frac{1}{\tan \left(z\right)}\dots z\in \mathbb{C}\\ & =& \frac{\cos \left(z\right)}{\sin \left(z\right)}\\ & =& \frac{\cos (x+iy)}{\sin (x+iy)}\dots x,y\in \mathbb{R}\\ & =& \frac{\cos \left(x\right)\cos \left(iy\right)-\sin \left(x\right)\sin \left(iy\right)}{\sin \left(x\right)\cos \left(iy\right)+\cos \left(x\right)\sin \left(iy\right)}\\ & & \dots \cos (\alpha +\beta )=\cos \left(\alpha \right)\cos \left(\beta \right)-\sin \left(\alpha \right)\sin \left(\beta \right)\\ & & \dots \sin (\alpha +\beta )=\cos \left(\alpha \right)\sin \left(\beta \right)+\sin \left(\alpha \right)\cos \left(\beta \right)\\ & =& \frac{\cos \left(x\right)\cosh \left(y\right)-\sin \left(x\right)i\sinh \left(y\right)}{\sin \left(x\right)\cosh \left(y\right)+\cos \left(x\right)i\sinh \left(y\right)}\\ & & \dots \cos \left(iy\right)=\frac{e^{i\left(iy\right)}+e^{-i\left(iy\right)}}{2}=\frac{e^{-y}+e^y}{2}=\cosh \left(y\right)\\ & & \dots \sin \left(iy\right)=\frac{e^{i\left(iy\right)}-e^{-i\left(iy\right)}}{2i}=\frac{e^{-y}-e^y}{2i}=\frac{1}{i}\frac{-(e^y-e^{-y})}{2}=\frac{i}{i}\frac{-1}{i}\sinh \left(y\right)=i\sinh \left(y\right)\\ & =& \frac{\cos \left(x\right)\cosh \left(y\right)-i\sin \left(x\right)\sinh \left(y\right)}{\sin \left(x\right)\cosh \left(y\right)+i\cos \left(x\right)\sinh \left(y\right)}\frac{\sin \left(x\right)\cosh \left(y\right)-i\cos \left(x\right)\sinh \left(y\right)}{\sin \left(x\right)\cosh \left(y\right)-i\cos \left(x\right)\sinh \left(y\right)}\\ & =& \frac{\cos \left(x\right)\cosh \left(y\right)\sin \left(x\right)\cosh \left(y\right)+\cos \left(x\right)\cosh \left(y\right)(-i\cos \left(x\right)\sinh \left(y\right))-i\sin \left(x\right)\sinh \left(y\right)\sin \left(x\right)\cosh \left(y\right)-i\sin \left(x\right)\sinh \left(y\right)(-i\cos \left(x\right)\sinh \left(y\right))}{{\sin }^2\left(x\right){\cosh }^2\left(y\right)+{\cos }^2\left(x\right){\sinh }^2\left(y\right)}\\ & =& \frac{\cos \left(x\right)\sin \left(x\right){\cosh }^2\left(y\right)-i{\cos }^2\left(x\right)\cosh \left(y\right)\sinh \left(y\right)-i{\sin }^2\left(x\right)\cosh \left(y\right)\sinh \left(y\right)-\cos \left(x\right)\sin \left(x\right)\sinh \left(y\right)}{{\sin }^2\left(x\right){\cosh }^2\left(y\right)+(1-{\sin }^2\left(x\right)){\sinh }^2\left(y\right)}\\ & =& \frac{\cos \left(x\right)\sin \left(x\right){\cosh }^2\left(y\right)-i{\cos }^2\left(x\right)\cosh \left(y\right)\sinh \left(y\right)-i{\sin }^2\left(x\right)\cosh \left(y\right)\sinh \left(y\right)-\cos \left(x\right)\sin \left(x\right)\sinh \left(y\right)}{{\sin }^2\left(x\right){\cosh }^2\left(y\right)+{\sinh }^2\left(y\right)-{\sin }^2\left(x\right){\sinh }^2\left(y\right)}\\ & =& \frac{\cos \left(x\right)\sin \left(x\right)({\cosh }^2\left(y\right)-\sinh \left(y\right))-i\{({\cos }^2\left(x\right)+{\sin }^2\left(x\right))\cosh \left(y\right)\sinh \left(y\right)\}}{{\sin }^2\left(x\right)({\cosh }^2\left(y\right)-{\sinh }^2\left(y\right))+{\sinh }^2\left(y\right)}\\ & =& \frac{\cos \left(x\right)\sin \left(x\right)-i\cosh \left(y\right)\sinh \left(y\right)}{{\sin }^2\left(x\right)+{\sinh }^2\left(y\right)}\\ & =& \frac{\cos \left(x\right)\sin \left(x\right)}{{\sin }^2\left(x\right)+{\sinh }^2\left(y\right)}+i\frac{-\cosh \left(y\right)\sinh \left(y\right)}{{\sin }^2\left(x\right)+{\sinh }^2\left(y\right)}\\ & =& u(x,y)+iv(x,y)\end{array}$$ $$\{\begin{array}{rcl}u(x,y)& =& \frac{\cos \left(x\right)\sin \left(x\right)}{{\sin }^2\left(x\right)+{\sinh }^2\left(y\right)}\\ v(x,y)& =& \frac{-\cosh \left(y\right)\sinh \left(y\right)}{{\sin }^2\left(x\right)+{\sinh }^2\left(y\right)}\end{array}$$