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 $z = x + iy$

Analyticity

$e^z$ is analytic on $\mathbb{C}$, with $\dfrac{\text{d}}{\text{d}z}\,e^z = e^z$.

Mapping

Let $\omega = e^z$ with $z = x + iy$. In polar form, $\omega$ has modulus $e^x$ and argument $y$. So the real part $x$ of $z$ sets the distance of $\omega$ from the origin, and the imaginary part $y$ sets the angle of $\omega$.

• A vertical line $x = c$ maps onto the circle $|\omega| = e^c$ centered at the origin. As $y$ increases by $2\pi$, the image goes around the circle once.
• A horizontal line $y = c$ maps onto the ray $\arg \omega = c$ from the origin, excluding the origin.
• A horizontal strip $\alpha < y < \alpha + 2\pi$ maps one-to-one onto the plane with the origin and the ray $\arg \omega = \alpha$ removed.
• A rectangle $a < x < b$, $\alpha < y < \beta$ with $\beta - \alpha \le 2\pi$ maps one-to-one onto a part of an annulus, between the circles $|\omega| = e^a$ and $|\omega| = e^b$ and between the rays $\arg \omega = \alpha$ and $\arg \omega = \beta$.

The whole $z$-plane maps onto $\mathbb{C} \setminus \{0\}$, covering each point infinitely many times, once per strip of height $2\pi$. The image never contains $0$, since $|e^z| = e^x > 0$.