10.5.39 Conformal mappings

Problem: Find the image of domain \[ D=\left\{|\operatorname{Re} z|<\frac{\pi}{2}\right\} \backslash\left\{z=t, \quad t \in\left[0 ; \frac{\pi}{2}\right)\right\} \] for the mapping \( f(z)=e^{i z} \).