# Definition:Exponential

## Contents

## Definition

The **exponential function** is denoted $\exp$ and can be defined in several ways, as described below.

## Real Numbers

For all definitions of the **real exponential function**:

- The domain of $\exp$ is $\R$

- The codomain of $\exp$ is $\R_{>0}$

For $x \in \R$, the real number $\exp x$ is called the **exponential of $x$**.

### As a Sum of a Series

The **exponential function** can be defined as a power series:

- $\exp x := \displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$

### As a Limit of a Sequence

The **exponential function** can be defined as the following limit of a sequence:

- $\exp x := \displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$

### As an Extension of the Rational Exponential

Let $e$ denote Euler's number.

Let $f: \Q \to \R$ denote the real-valued function defined as:

- $f \left({ x }\right) = e^x$

That is, let $f \left({ x }\right)$ denote $e$ to the power of $x$, for rational $x$.

Then $\exp : \R \to \R$ is defined to be the unique continuous extension of $f$ to $\R$.

$\exp \left({ x }\right)$ is called the **exponential of $x$**.

### As the Inverse to the Natural Logarithm

Consider the natural logarithm $\ln x$, which is defined on the open interval $\left({0 \,.\,.\, +\infty}\right)$.

From Logarithm is Strictly Increasing:

- $\ln x$ is strictly increasing.

From Inverse of Strictly Monotone Function:

- the inverse of $\ln x$ always exists.

The inverse of the natural logarithm function is called the **exponential function**, which is denoted as $\exp$.

Thus for $x \in \R$, we have:

- $y = \exp x \iff x = \ln y$

### As the Solution of a Differential Equation

The **exponential function** can be defined as the unique solution $y = f(x)$ to the first order ODE:

- $\dfrac{\mathrm d y}{\mathrm d x} = y$

...satisfying the initial condition $f(0) = 1$.

That is, the defining property of $\exp$ is that it is its own derivative.

## Complex Numbers

For all definitions of the **complex exponential function**:

- The domain of $\exp$ is $\C$.

- The image of $\exp$ is $\C \setminus \left\{ {0}\right\}$, as shown in Image of Complex Exponential Function.

For $z \in \C$, the complex number $\exp z$ is called the **exponential of $z$**.

### As a Sum of a Series

The **exponential function** can be defined as a power series:

- $\exp z := \displaystyle \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$

### By Real Functions

The **exponential function** can be defined by the real exponential, sine and cosine functions:

- $\exp z := e^x \left({\cos y + i \sin y}\right)$

where $z = x + iy$ with $x, y \in \R$.

Here, $e^x$ denotes the real exponential function, which must be defined beforehand.

### As a Limit of a Sequence

The **exponential function** can be defined as a limit of a sequence:

- $\displaystyle \exp z := \lim_{n \to \infty} \left({1 + \dfrac z n}\right)^n$

### As the Solution of a Differential Equation

The **exponential function** can be defined as the unique solution $y = f \left({z}\right)$ to the first order ODE:

- $\dfrac {\mathrm d y} {\mathrm d z} = y$

satisfying the initial condition $f \left({0}\right) = 1$.

That is, the defining property of $\exp$ is that it is its own derivative.

## Notation

The **exponential of $x$**, $\exp x$, is frequently written as $e^x$. The consistency of this power notation is demonstrated in Exp x equals e^x for Rational Numbers.

## Also see

## Linguistic Note

The word **exponential** derives ultimately from the (now archaic) verb **to expone**, which means **to set forth**, in the sense of **to expound**, or **explain**.

This itself comes from the Latin **expono**, meaning **I expose**, or **I exhibit**.

The word **exponent** (from which **exponential** is formed) therefore means **a person (or statement) that explains something**.

## Historical Note

The exponential function in its modern form is as a result of the original work done by Leonhard Paul Euler.

## Also see

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles - Weisstein, Eric W. "Exponential Function." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/ExponentialFunction.html