Complex Analysis: Correspodence between $H(\Omega_1)$ and $H(\Omega_2)$. If we let 
$$\Omega_1=\left\{z \in \mathbb{C} : 0<\operatorname{Im}(z)< \pi\}, \quad \Omega_2=\{z \in \mathbb{C}: 0<\operatorname{Im}(z) \right\},$$
can we establish a one-to-one correspondence between $H(\Omega_1)$ and $H(\Omega_2)$ where $H(\Omega)$ represents the set of real-valued harmonic functions on $\Omega \subset \mathbb{C}$?
Thoughts:
Can we appeal to simple conformal mapping and say that, if we can establish a conformal map between $\Omega_1$ and $\Omega_2$ then the correspondence exists as the harmonic nature of the functions is preserved under such a map i.e. the composition of a holomorphic and harmonic map is harmonic? If so, what could this conformal map be?
If we apply $z \mapsto e^z$ initially, this transforms the 'strip'to a 'wedge', but how do we advance from here?
If such a method is incorrect, how else can we demonstrate the existence of the correspondence?
Any help would be greatly appreciated. Best, MM.

The Riemann Mapping theorem says there is a differentiable map between these two sets that is compositionally invertible.  This just might do the job.
