The exponential converts vertical height to radius and horizontal position to angle, sending the half-strip to the right half-disc.Suggest a slogan
The map \[ F\colon\ts{z\st\abs{\Re z}<\tfrac\pi2,\ \Im z>0}\to\ts{w\st\abs{w}<1,\ \Re w>0},\qquad F(z)=e^{iz}, \] is a biholomorphism from the vertical half-strip onto the right half-disc, with inverse \(w\mapsto-i\Log w\), where \(\Log\) is the principal branch of the logarithm.
For \(z=x+iy\), \(e^{iz}=e^{-y}e^{ix}\). As \(y\) ranges over \((0,\infty)\) the modulus \(e^{-y}\) ranges bijectively over \((0,1)\), and as \(x\) ranges over \((-\pi/2,\pi/2)\) the argument \(x\) ranges over the arguments of points with positive real part. So \(F\) is a bijection onto the right half-disc. For such \(w=\rho e^{ix}\), \(-i\Log w=-i(\ln\rho+ix)=x-i\ln\rho=x+iy\) with \(y=-\ln\rho\), which is the holomorphic inverse.