Editing Exponential function (section)

{{Short description|Mathematical function, denoted exp(x) or e^x}}
{{About|the function {{math|{{var|f}}({{var|x}}) {{=}} {{var|e}}{{sup|{{var|x}}}}}} and its generalizations|functions of the form {{math|{{var|f}}({{var|x}}) {{=}} {{var|x}}{{sup|{{var|r}}}}}}|Power function|the bivariate function {{math|{{var|f}}({{var|x}},{{var|y}}) {{=}} {{var|x}}{{sup|{{var|y}}}}}}|Exponentiation|the representation of scientific numbers|E notation}}
{{Use dmy dates|date=August 2019|cs1-dates=y}}
{{Infobox mathematical function
| name = Exponential
| image = Image:exp.svg
| imagealt = Graph of the exponential function 
| caption = Graph of the exponential function
| general_definition = <math>\exp z = e^{z}</math>
| motivation_of_creation =
| fields_of_application =
| domain = <math>\mathbb{C}</math>
| range = <math>\begin{cases} (0,\infty) & \text{for }z \in \mathbb{R} \\ \mathbb{C} \setminus \{0\} & \text{for }z \in \mathbb{C} \end{cases}</math>
| zero = 1
| vr1 = 1
| f1 = [[Euler's number|{{math|''e''}}]]
| fixed = [[Lambert W function|{{math|−''W''{{sub|''n''}}(−1)}}]] for <math>n \in \mathbb{Z}</math>
| reciprocal = <math>\exp(-z)</math>
| inverse = [[Natural logarithm]], [[Complex logarithm]]
| derivative = <math>\exp'\! z = \exp z</math>
| antiderivative = <math>\int \exp z\,dz = \exp z + C</math>
| taylor_series = <math>\exp z = \sum_{n=0}^\infty\frac{z^n}{n!}</math>
}}

In [[mathematics]], the '''exponential function''' is the unique [[real function]] which maps [[0|zero]] to [[1|one]] and has a [[derivative (mathematics)|derivative]] everywhere equal to its value. The exponential of a variable {{tmath|x}} is denoted {{tmath|\exp x}} or {{tmath|e^x}}, with the two notations used interchangeably. It is called ''exponential'' because its argument can be seen as an [[exponent (mathematics)|exponent]] to which a constant [[e (mathematical constant)|number {{math|''e'' ≈ 2.718}}]], the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: it maps the [[additive identity]] {{math|0}} to the [[multiplicative identity]] {{math|1}}, and the exponential of a sum is equal to the product of separate exponentials, {{tmath|1=\exp(x + y) = \exp x \cdot \exp y }}. Its [[inverse function]], the [[natural logarithm]], {{tmath|\ln}} or {{tmath|\log}}, converts products to sums: {{tmath|1= \ln(x\cdot y) = \ln x + \ln y}}.

The exponential function is occasionally called the '''natural exponential function''', matching the name ''natural logarithm'', for distinguishing it from some other functions that are also commonly called ''exponential functions''. These functions include the functions of the form {{tmath|1=f(x) = b^x}}, which is [[exponentiation]] with a fixed base {{tmath|b}}. More generally, and especially in applications, functions of the general form {{tmath|1=f(x) = ab^x}} are also called exponential functions. They [[exponential growth|grow]] or [[exponential decay|decay]] exponentially in that the rate that {{tmath|f(x)}} changes when {{tmath|x}} is increased is ''proportional'' to the current value of {{tmath|f(x)}}.

The exponential function can be generalized to accept [[complex number]]s as arguments. This reveals relations between multiplication of complex numbers, rotations in the [[complex plane]], and [[trigonometry]]. [[Euler's formula]] {{tmath|1= \exp i\theta = \cos\theta + i\sin\theta}} expresses and summarizes these relations.

The exponential function can be even further generalized to accept other types of arguments, such as [[Matrix exponentiation|matrices]] and elements of [[Exponential map (Lie theory)|Lie algebras]].