Exponential of Sum

Theorem

Real Numbers

Let $x, y \in \R$ be real numbers.

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

Then:

$\map \exp {x + y} = \paren {\exp x} \paren {\exp y}$

Complex Numbers

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

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

Then:

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

Corollary

Real Numbers

Let $x, y \in \R$ be real numbers.

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

Then:

$\map \exp {x - y} = \dfrac {\exp x} {\exp y}$

Complex Numbers

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

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

Then:

$\map \exp {z_1 - z_2} = \dfrac {\exp z_1} {\exp z_2}$