I have two groups $(W = \{z \in \mathbb{C} \mid |z|=1\}, \cdot)$ and $(\mathbb{R},+)$ and the direct product $(W \times \mathbb{R}, *)$ where $*: (W \times \mathbb{R}) \times W (\times \mathbb{R}) \to (W \times \mathbb{R}) : ((a,b),(c,d)) \mapsto (a \cdot c, b + d)$.
My question now is how I could define an isomorphism $f: (W \times \mathbb{R}) \to \mathbb{C}_0$, where $\mathbb{C}_0 = \mathbb{C} \setminus {0}$.
I already read the explanation with this question, which is quite similar, but I don't seem to be really understanding the solution completely, I'm afraid. Or I just really have a different question. The main struggle is the sum in $\mathbb{R},+$.
Thanks in advance
Hint:
Any nonzero complex number can be written uniquely as $re^{i\theta}$, where $r>0$ is real. Rewrite $r$ as $e^{\ln r} = e^s$.