Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2010 / Analysis / 4 / iv

The complex exponential function $E(z)=e^z$ maps the strip bijectively and holomorphically to the upper half-plane: its modulus is $e^{\Re z}>0$ and its argument is $\Im z\in (0,\pi )$. Its inverse is the holomorphic logarithm $L(w)=\log |w|+i\mathrm{Arg}w$ with $\mathrm{Arg}w\in (0,\pi )$.

Part (iii) applies to both maps. The required inverse correspondences areThey compose to the identity because $L\circ E$ and $E\circ L$ do. This proves injectivity and surjectivity, not merely a map between the two sets.