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. Some old texts refer to the exponential function as the antilogarithm.^[1]