Definition:Exponential Function/Real
Definition
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 $\map f 0 = 1$.
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 Equivalence of Definitions of Real Exponential Function.
Also see
- Results about the exponential function can be found here.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.1$. Mappings: Example $45$