Definition:Exponential Function

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 $x$.

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

 * Basic Properties of Exponential Function

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.