Exponential of Sum/Complex Numbers/General Result/Corollary

Theorem

Let $m \in \Z_{>0}$ be a positive integer.

Let $z \in \C$ be a complex number.

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

Then:

$\ds \exp \paren {m z} = \paren {\exp z}^m$

Proof

From Exponential of Sum: Complex Numbers: General Result:

$\ds \exp \paren {\sum_{j \mathop = 1}^m z_j} = \prod_{j \mathop = 1}^m \paren {\exp z_j}$

for complex numberst $z_1, z_2, \ldots, z_m \in \C$.

The result follows by setting $z = z_1 = z_2 = \cdots = z_m$.

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations