# Definition:Exponential Function

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## 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 Power Series Expansion

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

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

### As a Limit of a Sequence

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

- $\exp x := \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$

### 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:

- $\map f x = e^x$

That is, let $\map f x$ 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 $\R$.

$\map \exp x$ 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 $\openint 0 {+\infty}$.

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 $y = \map f x$ to the first order ODE:

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

satisfying the initial condition $\map f 0 = 1$.

## Complex Numbers

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

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

- The image of $\exp$ is $\C \setminus \set 0$, 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 Power Series Expansion

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

\(\ds \forall z \in \C: \, \) | \(\ds \exp z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\) |

### By Real Functions

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

- $\exp z := e^x \paren {\cos y + i \sin y}$

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

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

### As a Limit of a Sequence

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

- $\ds \exp z := \lim_{n \mathop \to \infty} \paren {1 + \dfrac z n}^n$

### As the Solution of a Differential Equation

The **exponential function** can be defined as the unique particular solution $y = \map f z$ to the first order ODE:

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

satisfying the initial condition $\map f 0 = 1$.

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

## Notation

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

## Also see

- Equivalence of Definitions of Complex Exponential Function
- Properties of Exponential Function
- Properties of Complex Exponential Function

- Results about
**the exponential function**can be found**here**.

## Historical Note

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

## 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**.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles: The number $e$

- Weisstein, Eric W. "Exponential Function." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/ExponentialFunction.html