cot(z)をu(x,y)+iv(x,y)で表す

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