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The exponential function is a mathematical
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
denoted by f(x)=\exp(x) or e^x (where the argument is written as an
exponent Exponentiation is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
). It can be defined in several equivalent ways. Its ubiquitous occurrence in
pure Pure may refer to: Computing * A pure function * A virtual function, pure virtual function * PureSystems, a family of computer systems introduced by IBM in 2012 * Pure Software, a company founded in 1991 by Reed Hastings to support the Purify too ...
and
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...
has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". Its value at 1, e = \exp(1), is a
mathematical constant A mathematical constant is a key number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formal ...
called
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of a logarithm, base of the natural logarithm. It is the Limit of a sequence, limit of ...
. The exponential function equals its own
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
. Thus, it appears in the solutions of many
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
. Moreover, it satisfies the
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
:e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e=e^1, shows that e^n=\underbrace_ for positive integers , relating the exponential function to the elementary notion of
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
. The argument of the exponential function can be any
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
. This allows one to extend the concept of
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
(which would normally be defined only for integer exponents as: a^n=\underbrace_ for integer ) to
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
, by defining a^x:=\exp(x \ln(a)) for positive and real or complex . The exponential of a
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
argument is closely related to
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

trigonometry
as shown by
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

Euler's formula
. The argument can even be an entirely different kind of
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
(for example, a
square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
). In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing
population Population typically refers the number of people in a single area whether it be a city or town, region, country, or the world. Governments typically quantify the size of the resident population within their jurisdiction by a process called a ...
, a fund accruing compound
interest In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availa ...

interest
, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
,
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
,
chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in the real world. T ...

chemistry
,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
,
mathematical biology Mathematical and theoretical biology or, Biomathematics, is a branch of biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular bi ...
, and
economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interact ...

economics
. The real exponential function is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
from \mathbb to (0;\infty). Its
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
is the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
, denoted \ln, \log, or \log_e; because of this, some old texts refer to the exponential function as the antilogarithm. ''The'' exponential function is sometimes called the natural exponential function for distinguishing it from the other ''exponential functions'', which are the functions of the form f(x) = ab^x, where the base is a positive
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
. The definition b^x \ \ e^ for positive and
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
establishes a strong relationship between these functions, which explains this ambiguous terminology.


Graph

The
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
of y=e^x is upward-sloping, and increases faster as increases. The graph always lies above the -axis, but becomes arbitrarily close to it for large negative ; thus, the -axis is a horizontal
asymptote In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέ ...

asymptote
. The equation \tfrace^x = e^x means that the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

slope
of the
tangent In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
to the graph at each point is equal to its -coordinate at that point.


Relation to more general exponential functions

The exponential function f(x)=e^x is sometimes called the ''natural exponential function'' for distinguishing it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive , : ab^x := ae^ As functions of a real variable, exponential functions are uniquely characterized by the fact that the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
of such a function is
directly proportional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to the value of the function. The constant of proportionality of this relationship is the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
of the base : :\frac b^x \ \ \frac e^ = e^ \ln (b) = b^x \ln (b). For , the function b^x is increasing (as depicted for and ), because \ln b>0 makes the derivative always positive; while for , the function is decreasing (as depicted for ); and for the function is constant.
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of a logarithm, base of the natural logarithm. It is the Limit of a sequence, limit of ...
is the unique base for which the constant of proportionality is 1, since \ln(e)=1, so that the function is its own derivative: :\frac e^x = e^x \ln (e) = e^x. This function, also denoted as , is called the "natural exponential function", or simply "the exponential function". Since any exponential function can be written in terms of the natural exponential as b^x = e^, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by :x\mapsto e^x or x\mapsto \exp x. The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is a complicated expression. For real numbers and , a function of the form f(x)=ab^ is also an exponential function, since it can be rewritten as :ab^=\left(ab^d\right) \left(b^c\right)^x.


Formal definition

The real exponential function \exp\colon\mathbb\to\mathbb can be characterized in a variety of equivalent ways. It is commonly defined by the following
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
: :\exp x := \sum_^ \frac = 1 + x + \frac + \frac + \frac + \cdots Since the
radius of convergence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
of this power series is infinite, this definition is, in fact, applicable to all complex numbers z\in\mathbb (see for the extension of \exp x to the complex plane). The constant can then be defined as e=\exp 1=\sum_^\infty(1/k!). The term-by-term differentiation of this power series reveals that \frac\exp x = \exp x for all real , leading to another common characterization of \exp x as the unique solution of the
differential equation In mathematics, a differential equation is an equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

differential equation
:y'(x)=y(x), satisfying the initial condition y(0)=1. Based on this characterization, the chain rule shows that its inverse function, the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
, satisfies \frac\log_e y = 1/y for y>0, or \log_e y=\int_1^y \frac\,dt. This relationship leads to a less common definition of the real exponential function \exp x as the solution y to the equation : x = \int_1^y \frac \, dt. By way of the
binomial theorem In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...
and the power series definition, the exponential function can also be defined as the following limit: : \exp x = \lim_ \left(1 + \frac\right)^n. It can be shown that every
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
, nonzero solution of the functional equation f(x+y)=f(x)f(y) is an exponential function, f:\mathbb\to\mathbb,\ x\mapsto e^, with k\in\mathbb.


Overview

The exponential function arises whenever a quantity or at a rate
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
to its current value. One such situation is
continuously compounded interest Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then ea ...
, and in fact it was this observation that led
Jacob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) inclu ...
in 1683 to the number :\lim_\left(1 + \frac\right)^ now known as . Later, in 1697,
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss Swiss may refer to: * the adjectival form of Switzerland , french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Svizra , government_type = ...

Johann Bernoulli
studied the calculus of the exponential function. If a principal amount of 1 earns interest at an annual rate of compounded monthly, then the interest earned each month is times the current value, so each month the total value is multiplied by , and the value at the end of the year is . If instead interest is compounded daily, this becomes . Letting the number of time intervals per year grow without bound leads to the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

limit
definition of the exponential function, :\exp x = \lim_\left(1 + \frac\right)^ first given by
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Leonhard Euler
. This is one of a number of
characterizations of the exponential functionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
; others involve
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
or
differential equation In mathematics, a differential equation is an equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

differential equation
s. From any of these definitions it can be shown that the exponential function obeys the basic
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
identity, :\exp(x + y) = \exp x \cdot \exp y which justifies the notation for . The
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

derivative
(rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change ''proportional'' to the function itself (rather than equal to it) is expressible in terms of the exponential function. This function property leads to
exponential growth Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change ...

exponential growth
or
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative rela ...

exponential decay
. The exponential function extends to an
entire function In complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...
on the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
.
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

Euler's formula
relates its values at purely imaginary arguments to
trigonometric functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

trigonometric functions
. The exponential function also has analogues for which the argument is a
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
, or even an element of a
Banach algebra Banach is a Polish-language surname of several possible origins."Banach"
at genezanazwisk.pl (the webpage cites the sources)
or a
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
.


Derivatives and differential equations

The importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and is equal to 1 when . That is, : \frace^x = e^x \quad\text\quad e^0=1. Functions of the form for constant are the only functions that are equal to their derivative (by the
Picard–Lindelöf theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
). Other ways of saying the same thing include: * The slope of the graph at any point is the height of the function at that point. * The rate of increase of the function at is equal to the value of the function at . * The function solves the
differential equation In mathematics, a differential equation is an equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

differential equation
. * is a fixed point of derivative as a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) In architecture File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...
. If a variable's growth or decay rate is
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
to its size—as is the case in unlimited population growth (see
Malthusian catastrophe Malthusianism is the idea that population growth is potentially exponential while the growth of the food supply or other resources is linear growth, linear, which eventually reduces living standards to the point of triggering a depopulation, popula ...
), continuously compounded
interest In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money availa ...

interest
, or
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consi ...

radioactive decay
—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant , a function satisfies if and only if for some constant . The constant ''k'' is called the decay constant, disintegration constant, rate constant, or transformation constant. Furthermore, for any differentiable function , we find, by the chain rule: : \frac e^ = f'(x)e^.


Continued fractions for

A
continued fraction In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
for can be obtained via an identity of Euler: : e^x = 1 + \cfrac The following
generalized continued fractionIn complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates ...
for converges more quickly: : e^z = 1 + \cfrac or, by applying the substitution : : e^\frac = 1 + \cfrac with a special case for : : e^2 = 1 + \cfrac = 7 + \cfrac This formula also converges, though more slowly, for . For example: : e^3 = 1 + \cfrac = 13 + \cfrac


Complex plane

As in the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
case, the exponential function can be defined on the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
in several equivalent forms. The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: :\exp z := \sum_^\infty\frac Alternatively, the complex exponential function may defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one: :\exp z := \lim_\left(1+\frac\right)^n For the power series definition, term-wise multiplication of two copies of this power series in the
Cauchy Baron Augustin-Louis Cauchy (; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was ...
sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments: :\exp(w+z)=\exp w\exp z \text w,z\in\mathbb The definition of the complex exponential function in turn leads to the appropriate definitions extending the
trigonometric functions In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

trigonometric functions
to complex arguments. In particular, when ( real), the series definition yields the expansion :\exp(it) = \left( 1-\frac+\frac-\frac+\cdots \right) + i\left(t - \frac + \frac - \frac+\cdots\right). In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of and , respectively. This correspondence provides motivation for cosine and sine for all complex arguments in terms of \exp(\pm iz) and the equivalent power series: :\begin \cos z &:= \frac = \sum_^(-1)^k \frac ,\quad \text \\ \sin z &:= \frac =\sum_^(-1)^k\frac \end \text z\in\mathbb. The functions , , and so defined have infinite radii of convergence by the
ratio test In mathematics, the ratio test is a convergence tests, test (or "criterion") for the convergent series, convergence of a series (mathematics), series :\sum_^\infty a_n, where each term is a real number, real or complex number and is nonzero when ...
and are therefore
entire function In complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...
s (that is,
holomorphic Image:Conformal map.svg, A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics, a holomorphic function is a complex-valued function of one or more complex number, complex variables that is, at every point of ...
on \mathbb). The range of the exponential function is \mathbb\setminus \, while the ranges of the complex sine and cosine functions are both \mathbb in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of \mathbb, or \mathbb excluding one lacunary value. These definitions for the exponential and trigonometric functions lead trivially to
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

Euler's formula
: :\exp(iz)=\cos z+i\sin z \text z\in\mathbb. We could alternatively define the complex exponential function based on this relationship. If , where and are both real, then we could define its exponential as :\exp z = \exp(x+iy) := (\exp x)(\cos y + i \sin y) where , , and on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means. For t\in\mathbb , the relationship \overline=\exp(-it) holds, so that , \exp(it), =1 for real t and t\mapsto\exp(it) maps the real line (mod ) to the
unit circle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

unit circle
in the complex plane. Moreover, going from t=0 to t=t_0, the curve defined by \gamma(t)=\exp(it) traces a segment of the unit circle of length :\int_0^, \gamma'(t), dt = \int_0^ , i\exp(it), dt=t_0, starting from in the complex plane and going counterclockwise. Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions. The complex exponential function is periodic with period and \exp(z+2\pi i k)=\exp z holds for all z\in\mathbb,k\in\mathbb. When its domain is extended from the real line to the complex plane, the exponential function retains the following properties: :\begin e^ = e^z e^w\, \\ e^0 = 1\, \\ e^z \ne 0 \\ \tfrac e^z = e^z \\ \left(e^z\right)^n = e^, n \in \mathbb \end \text w,z\in\mathbb. Extending the natural logarithm to complex arguments yields the
complex logarithm of the color is used to show the ''arg Arg or ARG may refer to: Places *''Arg'' () means "citadel" in Persian, and may refer to: **Arg, Iran, a village in Fars Province, Iran **Arg (Kabul), presidential palace in Kabul, Afghanistan **Arg, South ...
, which is a
multivalued function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

multivalued function
. We can then define a more general exponentiation: : z^w = e^ for all complex numbers and . This is also a multivalued function, even when is real. This distinction is problematic, as the multivalued functions and are easily confused with their single-valued equivalents when substituting a real number for . The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context: : , but rather multivalued over integers See failure of power and logarithm identities for more about problems with combining powers. The exponential function maps any
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...
in the complex plane to a
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar __NOTOC__ has an infinitely repeating self-similarity when it is magnified. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of s ...

logarithmic spiral
in the complex plane with the center at the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
. Two special cases exist: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius. Image:ExponentialAbs_real_SVG.svg, Image:ExponentialAbs_image_SVG.svg, Image:ExponentialAbs_SVG.svg, Considering the complex exponential function as a function involving four real variables: : v+iw = \exp(x+iy) the graph of the exponential function is a two-dimensional surface curving through four dimensions. Starting with a color-coded portion of the xy domain, the following are depictions of the graph as variously projected into two or three dimensions. File: Complex exponential function graph domain xy dimensions.svg, Checker board key:
x> 0:\; \text
x< 0:\; \text
y> 0:\; \text
y< 0:\; \text File: Complex exponential function graph range vw dimensions.svg, Projection onto the range complex plane (V/W). Compare to the next, perspective picture. File: Complex exponential function graph horn shape xvw dimensions.jpg, Projection into the x, v, and w dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image). File: Complex exponential function graph spiral shape yvw dimensions.jpg, Projection into the y, v, and w dimensions, producing a spiral shape. (y range extended to ±2, again as 2-D perspective image).
The second image shows how the domain complex plane is mapped into the range complex plane: * zero is mapped to 1 * the real x axis is mapped to the positive real v axis * the imaginary y axis is wrapped around the unit circle at a constant angular rate * values with negative real parts are mapped inside the unit circle * values with positive real parts are mapped outside of the unit circle * values with a constant real part are mapped to circles centered at zero * values with a constant imaginary part are mapped to rays extending from zero The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image. The third image shows the graph extended along the real x axis. It shows the graph is a surface of revolution about the x axis of the graph of the real exponential function, producing a horn or funnel shape. The fourth image shows the graph extended along the imaginary y axis. It shows that the graph's surface for positive and negative y values doesn't really meet along the negative real v axis, but instead forms a spiral surface about the y axis. Because its y values have been extended to , this image also better depicts the 2π periodicity in the imaginary y value.


Computation of where both and are complex

Complex exponentiation can be defined by converting to polar coordinates and using the identity : : a^b = \left(re^\right)^b = \left(e^\right)^b = e^ However, when is not an integer, this function is , because is not unique (see failure of power and logarithm identities).


Matrices and Banach algebras

The power series definition of the exponential function makes sense for square
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
(for which the function is called the
matrix exponential In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
) and more generally in any unital
Banach algebra Banach is a Polish-language surname of several possible origins."Banach"
at genezanazwisk.pl (the webpage cites the sources)
. In this setting, , and is invertible with inverse for any in . If , then , but this identity can fail for noncommuting and . Some alternative definitions lead to the same function. For instance, can be defined as :\lim_ \left(1 + \frac \right)^n . Or can be defined as , where is the solution to the differential equation , with initial condition ; it follows that for every in


Lie algebras

Given a
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
and its associated
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
\mathfrak, the exponential map is a map \mathfrak satisfying similar properties. In fact, since is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group of invertible matrices has as Lie algebra , the space of all matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map. The identity can fail for Lie algebra elements and that do not commute; the
Baker–Campbell–Hausdorff formula In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z to the equation :e^X e^Y=e^Z for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield ...
supplies the necessary correction terms.


Transcendency

The function is not in (that is, is not the quotient of two polynomials with complex coefficients). For distinct complex numbers , the set is linearly independent over . The function is transcendental over .


Computation

When computing (an approximation of) the exponential function near the argument , the result will be close to 1, and computing the value of the difference \exp x-1 with
floating-point arithmetic In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
may lead to the loss of (possibly all)
significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are Numerical digit, digits in the number that are reliable and absolutely necessary to indicate the quantity of som ...
, producing a large calculation error, possibly even a meaningless result. Following a proposal by
William Kahan William "Velvel" Morton Kahan (born June 5, 1933) is a Canadian Canadians (french: Canadiens) are people identified with the country of Canada. This connection may be residential, legal, historical or cultural. For most Canadians, many (or al ...

William Kahan
, it may thus be useful to have a dedicated routine, often called expm1, for computing directly, bypassing computation of . For example, if the exponential is computed by using its
Taylor series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
:e^x=1+x+\frac 2 + \frac6+\cdots +\frac+\cdots, one may use the Taylor series of e^x-1: :e^x-1=x+\frac 2 + \frac6+\cdots +\frac+\cdots. This was first implemented in 1979 in the
Hewlett-Packard The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company headquartered in Palo Alto, California Palo Alto (; Spanish language, Spanish for "tall stick" ...

Hewlett-Packard
HP-41C The HP-41C series are programmable, expandable, continuous memoryThe term continuous memory was coined by Hewlett-Packard (HP) to describe a unique feature of certain HP calculators Image:HP48G.jpg, 200px, HP 48G HP calculators are various calc ...

HP-41C
calculator, and provided by several calculators,
operating system An operating system (OS) is system software System software is software designed to provide a platform for other software. Examples of system software include operating systems (OS) like macOS, Linux, Android (operating system), Android and Mi ...

operating system
s (for example Berkeley UNIX 4.3BSD), computer algebra systems, and programming languages (for example C99). In addition to base , the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: 2^x - 1 and 10^x - 1. A similar approach has been used for the logarithm (see lnp1). An identity in terms of the hyperbolic tangent, :\operatorname (x) = e^x - 1 = \frac, gives a high-precision value for small values of on systems that do not implement .


See also

* Carlitz exponential, a characteristic analogue * * * Gaussian function * Half-exponential function, a compositional square root of an exponential function * List of exponential topics * List of integrals of exponential functions * Mittag-Leffler function, a generalization of the exponential function * p-adic exponential function, -adic exponential function * Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions *


Notes


References

* * *


External links

* {{Authority control Elementary special functions Analytic functions Exponentials Special hypergeometric functions E (mathematical constant)