Exponential function

Exponential
Graph of the exponential function
General information
General definition	${\displaystyle \exp z=e^{z}}$
Domain, codomain and image
Domain	${\displaystyle \mathbb {C} }$
Image	${\displaystyle {\begin{cases}(0,\infty )&{\text{for }}z\in \mathbb {R} \\\mathbb {C} \setminus \{0\}&{\text{for }}z\in \mathbb {C} \end{cases}}}$
Specific values
At zero	1
Value at 1	e
Specific features
Fixed point	−Wn(−1) for ${\displaystyle n\in \mathbb {Z} }$
Related functions
Reciprocal	${\displaystyle \exp(-z)}$
Inverse	Natural logarithm, Complex logarithm
Derivative	${\displaystyle \exp '\!z=\exp z}$
Antiderivative	${\displaystyle \int \exp z\,dz=\exp z+C}$
Series definition
Taylor series	${\displaystyle \exp z=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}}$

In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. The exponential of a variable ⁠${\displaystyle x}$⁠ is denoted ⁠${\displaystyle \exp x}$⁠ or ⁠${\displaystyle e^{x}}$⁠, with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant number 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 0 to the multiplicative identity 1, and the exponential of a sum is equal to the product of separate exponentials, ⁠${\displaystyle \exp(x+y)=\exp x\cdot \exp y}$⁠. Its inverse function, the natural logarithm, ⁠${\displaystyle \ln }$⁠ or ⁠${\displaystyle \log }$⁠, converts products to sums: ⁠${\displaystyle \ln(x\cdot y)=\ln x+\ln y}$⁠.

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 ⁠${\displaystyle f(x)=b^{x}}$⁠, which is exponentiation with a fixed base ⁠${\displaystyle b}$⁠. More generally, and especially in applications, functions of the general form ⁠${\displaystyle f(x)=ab^{x}}$⁠ are also called exponential functions. They grow or decay exponentially in that the amount that ⁠${\displaystyle f(x)}$⁠ changes when ⁠${\displaystyle x}$⁠ is increased is proportional to the current value of ⁠${\displaystyle f(x)}$⁠.

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

The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras.