Exponentials

Overview

The natural exponential function refers to a function that applies exponentiation with Euler's number $e$ as the base. That is, a function $f(x)$ of form $f(x)=e^x$ for some $x\in \mathbb{R}$. This is alternatively denoted as $f(x)=\exp x$.

More generally, an exponential function refers to any function of form $f(x)=b^x$ where $b,x\in \mathbb{R}$. and $b>0$. General exponential functions are defined in terms of natural exponential functions via the following identity: $$b^x = e^{x\ln{b}}$$

Complex

If $z=x+iy$, we define $e^z$ to be the complex number given by $$e^z = e^x(\cos{y} + i\sin{y}).$$

Every complex number $z\ne 0$ can be expressed in the form $z=r{e}^{i\theta }$ where $r=|z|$ and $\theta =\text{Arg}(z)+2\pi n$ for any $n\in \mathbb{Z}$. This representation is called the polar form of $z$.

Multiplication

Let $w=r(\cos \alpha +i\sin \alpha )$ and $z=s(\cos \beta +i\sin \beta )$. Then $$wz = rs[\cos{(\alpha + \beta)} + i\sin{(\alpha + \beta)}].$$

Division

Let $w=r(\cos \alpha +i\sin \beta )$ and $z=s(\cos \beta +i\sin \beta )$. If $z\ne 0$, then $$\frac{w}{z} = \frac{r}{s}[\cos{(\alpha - \beta)} + i\sin{(\alpha - \beta)}].$$

De Moivre's Theorem

Let $z=r(\cos \theta +i\sin \theta )$ be a complex number and $n$ be any integer. Then $$z^n = r^n\left[ \cos{(n\theta)} + i\sin{(n\theta)} \right].$$

Roots

Let $z=r(\cos \theta +i\sin \theta )$ be a complex number. Then, for $k=0,1,\dots ,n-1$, the $n$th roots of $z$ are given by $$\large \sqrt[n]{r} \left[ \cos{\left( \frac{\theta + 2\pi k}{n} \right)} + i\sin{\left( \frac{\theta + 2\pi k}{n} \right)} \right].$$

The solutions to $x^n=1$ are called the $n$th roots of unity.

Bibliography

• Ted Sundstrom and Steven Schlicker, Trigonometry, 2024.
• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
• Wikipedia. “Exponential function.” January 3, 2026. https://en.wikipedia.org/w/index.php?title=Exponential_function.