Sum of Complex Indices of Real Number

Theorem
Let $r \in \R_{> 0}$ be a (strictly) positive real number.

Let $\psi, \tau \in \C$ be complex numbers.

Let $r^\lambda$ be defined as the the principal branch of a positive real number raised to a complex number.

Then:


 * $r^{\psi \mathop + \tau} = r^\psi \times r^\tau$

Proof
Then: