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:


 * $\map \exp {z_1 + z_2} = \paren {\exp z_1} \paren {\exp z_2}$

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


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

which satisfies the initial condition $\map \exp 0 = 1$.

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


 * $\map f z = \map \exp z \, \map \exp {z_1 + z_2 - z}$

Then find its derivative:

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

Then: