📂Functions

Exponential Functions

Overview

The Exponential Function is a generalization of exponentiation that appears universally across all branches of mathematics. Although in original exponentiations the base $a > 0$ does not necessarily have to be $a = e$, the existence of the base change formula means that essentially, it doesn’t matter which base is used. For convenience, when referring to an exponential function, its base is commonly considered as $e$.

Definition ¹

When $x, y \in \mathbb{R}$, for a complex number $z \in \mathbb{C}$, $\exp : \mathbb{C} \to \mathbb{C}$ is defined as follows: $$ \exp z = e^{z} = e^{x + iy} = e^{x} \left( \cos y + i \sin y \right) $$

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• $e = 2.71828182 \cdots$ is the Euler’s constant.

Derivation

This has been equally covered in the curriculum, but I’ll describe it with slightly more difficult terminology. For convenience, the base is unified as $e$, and I’ll explain with a review sentiment rather than convincing in detail to the level of high school students.

Natural Numbers $\mathbb{N}$

The very foundation of the exponential function, its exponentiation, can naturally be expressed for some natural number $n \in \mathbb{N}$ as follows. Here, $n$ used as a superscript to $e$ intuitively represents how many times $e$ has been multiplied. $$ e^{n} = \overbrace{e \cdot e \cdots e}^{n \text{ Times}} $$ Furthermore, for two natural numbers $n, m \in \mathbb{N}$, the following can easily be confirmed: $$ e^{n+m} = e^{n} e^{m} $$

Integer $\mathbb{Z}$

According to the Field Axioms, for all real numbers $a \ne 0$, an inverse element $a^{-1}$ for multiplication must exist. In other words, for all $a = e^{n}$, there exists an $a^{-1}$ that satisfies the following: $$ 1 = a \cdot a^{-1} = e^{n} \cdot a^{-1} $$ Intuitively, this corresponds to how many times $e$ is divided. Representing this inverse as $a^{-1} = e^{-n}$, the exponential function is thereby extended to all integers.

Rational Number $\mathbb{Q}$

Consider $e^{n}$ that satisfies the following for two integers $n,m \in \mathbb{Z}$ and $a^{m}$: $$ a^{m} = e^{n} $$ This means that squaring $e$ by $n$ times results in $a^{m}$. Now, if we denote it as $a = e^{ {{ n } \over { m }} }$, it can be expressed as: $$ a^{m} = \left( e^{ {{ n } \over { m }} } \right)^{m} = \overbrace{e^{ {{ n } \over { m }} } \cdots e^{ {{ n } \over { m }} }}^{m \text{ Times}} $$ Thus, it’s clear that the exponential function extends well to rational numbers.

Real Number $\mathbb{R}$

Due to the density of real numbers, there must exist a sequence of rational numbers $\left\{ r_{k} \right\}_{k=1}^{\infty}$ that converges to every real number $x \in \mathbb{R}$. Accordingly, the exponentiation of $e$ for the real number $x \in \mathbb{R}$ is defined as follows: $$ \exp(x) = e^{x} := \lim_{k \to \infty} e^{r_{k}} $$

Complex Number $\mathbb{C}$

Polar Representation of Complex Numbers: A complex number $z \ne 0$ corresponds to point $P(x,y)$ on the complex plane, and can be Polar Represented through the length of segment $\overline{OP}$, $r := |z|$, and the counterclockwise angle $\theta$ made with the $x$ axis as follows: $$ z = r \left( \cos \theta + i \sin \theta \right) $$

Finally, the extension to complex functions occurs formally as above. From the citation above, $$ r = e^{x} \\ e^{iy} = \cos y + i \sin y $$ is naturally obtained, and thus it is defined as an exponential function, a type of complex function. $$ \exp z = e^{z} = e^{x + iy} = e^{x} \left( \cos y + i \sin y \right) $$