Exponential Function

For any complex number $z$,

$\exp$ is an entire function.

For $z = x + iy$,

Properties

• $\dfrac{\text{d}}{\text{d}z}\,e^z = e^z$
• $e^{z_1 + z_2} = e^{z_1}e^{z_2}$
• $e^{i\theta} = \cos\theta + i\sin\theta$, for real $\theta$
• $e^{z + 2n\pi i} = e^z$, for all $z \in \mathbb{C}$ and $n \in \mathbb{Z}$
• $e^z = 1$ iff $z = 2n\pi i$, where $n \in \mathbb{Z}$
• $e^{z_1} = e^{z_2}$ iff $z_1 = z_2 + 2n\pi i$, where $n \in \mathbb{Z}$
• $|e^z| = e^x = e^{\operatorname{Re} z}$, for $z = x + iy$