Product of exponentials

Statement

For all complex numbers $z$ and $w$, $$e^{z + w} = e^{z} e^{w},$$ where $e^{z} = \sum_{n=0}^{\infty} z^n / n!$ is the complex exponential function.

Remarks

The proof multiplies the two series and uses the binomial theorem. It follows that $e^{z} \neq 0$ and $e^{-z} = 1 / e^{z}$ for each $z$. With Euler's formula, it gives the addition formulas for $\cos$ and $\sin$.

Explanations

L1 human

The exponential function turns sums into products, for complex numbers as for real numbers. For example, $e^{i \theta} e^{i \varphi} = e^{i (\theta + \varphi)}$: turning by $\theta$ and then by $\varphi$ is turning by $\theta + \varphi$.