Definition:Exponential Function

From ProofWiki
Jump to navigation Jump to search


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:

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


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:

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

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 $\openint 0 {+\infty}$.

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 $y = f \left({x}\right)$ to the first order ODE:

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

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


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 (complex) power series:

\(\displaystyle \exp z\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots\)


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:

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


As the Solution of a Differential Equation

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

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

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

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 $e^x$.


Also see

  • 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