Exponential of Sum
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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:
- $\map \exp {z_1 - z_2} = \dfrac {\exp z_1} {\exp z_2}$