Exponential of Sum/Complex Numbers

Theorem
Let $z_1, z_2 \in \C$ be complex numbers.

Let $\exp z$ be the exponential of $z$.

Then:


 * $\exp \left({z_1 + z_2}\right) = \left({\exp z_1}\right) \left({\exp z_2}\right)$

Proof
This proof is based on the definition of the complex exponential as the unique solution of the differential equation:


 * $\dfrac{\mathrm d}{\mathrm d z} \exp = \exp$

which satisfies the initial condition $\exp \left({0}\right) = 1$.

Define the complex function $f: \C \to \C$ by:


 * $f \left({z}\right) = \exp \left({z}\right) \exp \left({z_1 + z_2 - z}\right)$

Then find its derivative:

From Zero Derivative implies Constant Complex Function, it follows that $f$ is constant.

Then: