Definition:Exponential
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Definition
For all definitions of the exponential:
- The domain of $\exp$ is $\R$
- The codomain of $\exp$ is $\R_{>0}$
The scope of this needs to be expanded for the complex plane. This will not be possible till the basics of complex analysis have been properly addressed. See the talk page.
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Inverse of the Logarithm
Consider the natural logarithm $\ln x$, which is defined on the open interval $\left({0 \,.\,.\, \infty}\right)$.
From Logarithm is Strictly Increasing and Concave, $\ln x$ 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 and is written $\exp$.
Thus we have:
- $y = \exp x \iff x = \ln y$
The number $\exp x$ is called the exponential of $x$.
In Terms of Euler's Number
From the definition of powers for real numbers, we have:
- $z^x = \exp \left({x \ln z}\right)$
Suppose $z = e$, where $e$ is Euler's number, i.e. $2.71828\ldots$
From that definition of $e$, we have $\ln e = 1$.
Thus:
- $e^x = \exp \left({x \ln e}\right) = \exp x$
Thus $\exp x$ can be (and frequently is) written and defined as $e^x$.
So the number $e^x$ is also called the exponential of $x$ and the operation of raising $e$ to the power of $x$ is known as the exponential function.
As a Limit of a Sequence
The exponential function can also be defined as the following limit of a sequence:
- $e^x := \displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$
As a Sum of a Series
It can be defined as a power series:
- $e^x := \displaystyle \sum_{n = 0}^\infty \frac {x^n} {n!}$
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
The definition still holds when $x \in \C$ is a complex number.
Equivalence of Definitions
As shown in Equivalence of Definitions of Exponential the definitions above are equivalent.
Also see
- Results about the exponential function can be found here.
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
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 14.4
- Weisstein, Eric W. "Exponential Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExponentialFunction.html
