The function $f(z)=\exp(\exp(z))$ on the domain $U=\{z=x+iy:-\pi/2
$ |f(x\pm i\pi/2)|=\left|\exp(\exp(x\pm i\pi/2))\right| $ I don't know why it is 1?
Based on Harald's hint, I want to prove $e^{x\pm i\pi/2}$ is imaginary. $ e^{x\pm i\pi/2}=e^x[\cos(\pi/2)\pm i\sin(\pi/2)]=\pm ie^x. $