:PROPERTIES: :ID: ab249b72-fa90-4826-b9e4-c1e7d967ac82 :mtime: 20231026013503 :ctime: 20231026013502 :END: #+title: complex trig functions #+filetags: :public:project: * Example :$cos(z) = 2$ Let $z = x + iy$ \begin{align} &cos(z)\\ &= cos(x + iy) \\ &= cos(x)cos(iy) - sin(x)sin(iy)\\ &= cos(x)cosh(y) -isin(x)sinh(y)\\ \end{align} So \begin{align} &-isin(x)sinh(y) = 0\\ &\iff sin(x) = 0 \lor sinh(y) = 0 \\ &iff x = n\pi \lor y = 0 \end{align} \begin{align} & cos(x)cosh(y) = 2 \\ \implies cos(n\pi) = (-1)^{n} \end{align} so \[z = 2m\pi \pm i arccos(2)\] \[z = 2m\pi \pm i \log(2 + \sqrt{3})\]