Exponential function

Exponential
The natural exponential function along part of the real axis
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: ⁠${\displaystyle \ln(x\cdot y)=\ln x+\ln y}$⁠.

Other functions of the general form ⁠${\displaystyle f(x)=b^{x}}$⁠, with base ⁠${\displaystyle b}$⁠, are also commonly called exponential functions, and share the property of converting addition to multiplication, ⁠${\displaystyle b^{x+y}=b^{x}\cdot b^{y}}$⁠. Where these two meanings might be confused, the exponential function of base ⁠${\displaystyle e}$⁠ is occasionally called the natural exponential function, matching the name natural logarithm. The generalization of the standard exponent notation ⁠${\displaystyle b^{x}}$⁠ to arbitrary real numbers as exponents, is usually formally defined in terms of the exponential and natural logarithm functions, as ⁠${\displaystyle b^{x}=\exp(x\cdot \ln b)}$⁠. The "natural" base ⁠${\displaystyle e=\exp 1}$⁠ is the unique base satisfying the criterion that the exponential function's derivative equals its value, ⁠${\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}$⁠, which simplifies definitions and eliminates extraneous constants when using exponential functions in calculus.

Quantities which change over time in proportion to their value, for example the balance of a bank account bearing compound interest, the size of a bacterial population, the temperature of an object relative to its environment, or the amount of a radioactive substance, can be modeled using functions of the form ⁠${\displaystyle y(t)=ae^{ct}}$⁠, also sometimes called exponential functions; these quantities undergo exponential growth if ⁠${\displaystyle c}$⁠ is positive or exponential decay if ⁠${\displaystyle c}$⁠ is negative.

The exponential function can be generalized to accept a complex number as its argument. This reveals a relation between the multiplication of complex numbers and rotation in the Euclidean plane, Euler's formula ⁠${\displaystyle \exp i\theta =\cos \theta +i\sin \theta }$⁠: the exponential of an imaginary number ⁠${\displaystyle i\theta }$⁠ is a point on the complex unit circle at angle ⁠${\displaystyle \theta }$⁠ from the real axis. The identities of trigonometry can thus be translated into identities involving exponentials of imaginary quantities. The complex function ⁠${\displaystyle z\mapsto \exp z}$⁠ is a conformal map from an infinite strip of the complex plane ⁠${\displaystyle -\pi <\operatorname {Im} z\leq \pi }$⁠ (which periodically repeats in the imaginary direction) onto the whole complex plane except for ⁠${\displaystyle 0}$⁠.

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