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:

Radicals

A radical refers to an $n$th root of $x$ denoted as $\sqrt[n]{x}$. The positive integer $n$ is called the index or degree. The number $x$ of which the root is taken is the radicand.

Rationalization is a process by which radicals are eliminated from an expression. An expression of form $\sqrt{a}\pm \sqrt{b}$ is rationalized by multiplying by its conjugate $\sqrt{a}\mp \sqrt{b}$.

Complex

If $z=x+iy$, we define $e^z$ to be the complex number given by

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

Euler's formula is a specialization of the above definition where the real portion of the complex number is $0$:

Multiplication

Let $w=r(\cos \alpha +i\sin \alpha )$ and $z=s(\cos \beta +i\sin \beta )$. Then

Division

Let $w=r(\cos \alpha +i\sin \beta )$ and $z=s(\cos \beta +i\sin \beta )$. If $z\ne 0$, then

De Moivre's Theorem

Let $z=r(\cos \theta +i\sin \theta )$ be a complex number and $n$ be any integer. Then

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

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