The exponential function is a mathematical
function denoted by
or
(where the argument is written as an
exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
(repeated multiplication), but modern definitions (there are
several equivalent characterizations) allow it to be rigorously extended to all real arguments, including
irrational numbers. Its ubiquitous occurrence in
pure and
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
led mathematician
Walter Rudin to opine that the exponential function is "the most important function in mathematics".
The exponential function satisfies the exponentiation
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
which, along with the definition
, shows that
for positive integers , and relates the exponential function to the elementary notion of exponentiation. The base of the exponential function, its value at 1,
, is a ubiquitous
mathematical constant called
Euler's number.
While other continuous nonzero functions
that satisfy the exponentiation identity are also known as ''exponential functions'', the exponential function exp is the unique real-valued function of a real variable whose derivative is itself and whose value at is ; that is,
for all real , and
Thus, exp is sometimes called the natural exponential function to distinguish it from these other exponential functions, which are the functions of the form
where the base is a positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
. The relation
for positive and
real or
complex establishes a strong relationship between these functions, which explains this ambiguous terminology.
The real exponential function can also be defined as a power series. This power series definition is readily extended to complex arguments to allow the complex exponential function
to be defined. The complex exponential function takes on all complex values except for 0 and is closely related to the complex
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
, as shown by
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
.
Motivated by more abstract properties and characterizations of the exponential function, the exponential can be generalized to and defined for entirely different kinds of
mathematical objects (for example, a
square matrix or a
Lie algebra).
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 to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction usi ...
, a fund accruing compound
interest
In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distin ...
, 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 fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
,
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
,
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 ...
,
mathematical biology, and
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
.
The real exponential function is a
bijection from
to
. Its
inverse function is the
natural logarithm, denoted
or
because of this, some old texts
refer to the exponential function as the antilogarithm.
Graph
The
graph of
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. The equation
means that the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the
tangent to the graph at each point is equal to its -coordinate at that point.
Relation to more general exponential functions
The exponential function
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 ,
As functions of a real variable, exponential functions are uniquely
characterized by the fact that the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of such a function is
directly proportional to the value of the function. The constant of proportionality of this relationship is the
natural logarithm of the base :
For , the function
is increasing (as depicted for and ), because
makes the derivative always positive; while for , the function is decreasing (as depicted for ); and for the function is constant.
Euler's number is the unique base for which the constant of proportionality is 1, since
, so that the function is its own derivative:
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
, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by
or
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
is also an exponential function, since it can be rewritten as
Formal definition
The real exponential function
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 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
:
Since the
radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers
(see for the extension of
to the complex plane). The constant can then be defined as
The term-by-term differentiation of this power series reveals that
for all real , leading to another common characterization of
as the unique solution of the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
satisfying the initial condition
Based on this characterization, the
chain rule shows that its inverse function, the
natural logarithm, satisfies
for
or
This relationship leads to a less common definition of the real exponential function
as the solution
to the equation
By way of the
binomial theorem and the power series definition, the exponential function can also be defined as the following limit:
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 g ...
, nonzero solution of the functional equation
is an exponential function,
with
Overview
The exponential function arises whenever a quantity
grows or
decays at a rate
proportional to its current value. One such situation is
continuously compounded interest, and in fact it was this observation that led
Jacob Bernoulli in 1683
to the number
now known as . Later, in 1697,
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'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
definition of the exponential function,
first given by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
.
This is one of a number of
characterizations of the exponential function; others involve
series or
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s.
From any of these definitions it can be shown that the exponential function obeys the basic
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
identity,
which justifies the notation for .
The
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
(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 or
exponential decay.
The exponential function extends to an
entire function on the
complex plane.
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
relates its values at purely imaginary arguments to
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
. The exponential function also has analogues for which the argument is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, or even an element of a
Banach algebra or a
Lie algebra.
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,
Functions of the form for constant are the only functions that are equal to their derivative (by the
Picard–Lindelöf theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cau ...
). 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 that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
.
* is a
fixed point of derivative as a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
.
If a variable's growth or decay rate is
proportional 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, which eventually reduces living standards to the point of triggering a population die off. This event, c ...
), continuously compounded
interest
In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distin ...
, or
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:
Continued fractions for
A
continued fraction for can be obtained via
an identity of Euler:
The following
generalized continued fraction for converges more quickly:
or, by applying the substitution :
with a special case for :
This formula also converges, though more slowly, for . For example:
Complex plane
As in the
real case, the exponential function can be defined on the
complex plane 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:
Alternatively, the complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one:
For the power series definition, term-wise multiplication of two copies of this power series in the
Cauchy sense, permitted by
Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:
The definition of the complex exponential function in turn leads to the appropriate definitions extending the
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
to complex arguments.
In particular, when ( real), the series definition yields the expansion
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
and the equivalent power series:
for all
The functions , , and so defined have infinite
radii of convergence by the
ratio test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series
:\sum_^\infty a_n,
where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert ...
and are therefore
entire functions (that is,
holomorphic on
). The range of the exponential function is
, while the ranges of the complex sine and cosine functions are both
in its entirety, in accord with
Picard's theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard.
The theorems
Little Picard Theorem: If a function f: \mathbb \to\mathbb i ...
, which asserts that the range of a nonconstant entire function is either all of
, or
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 mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
:
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
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
, the relationship
holds, so that
for real
and
maps the real line (mod ) to the
unit circle in the complex plane. Moreover, going from
to
, the curve defined by
traces a segment of the unit circle of length
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
holds for all
.
When its domain is extended from the real line to the complex plane, the exponential function retains the following properties:
for all
Extending the natural logarithm to complex arguments yields the
complex logarithm , which is a
multivalued function.
We can then define a more general exponentiation:
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:
See
failure of power and logarithm identities for more about problems with combining powers.
The exponential function maps any
line in the complex plane to a
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:
the graph of the exponential function is a two-dimensional surface curving through four dimensions.
Starting with a color-coded portion of the
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:
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 , , and 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 , , and dimensions, producing a spiral shape. ( 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
axis is mapped to the positive real
axis
* the imaginary
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
axis. It shows the graph is a surface of revolution about the
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
axis. It shows that the graph's surface for positive and negative
values doesn't really meet along the negative real
axis, but instead forms a spiral surface about the
axis. Because its
values have been extended to , this image also better depicts the 2π periodicity in the imaginary
value.
Computation of where both and are complex
Complex exponentiation can be defined by converting to polar coordinates and using the identity :
However, when is not an integer, this function is
multivalued, 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 (for which the function is called the
matrix exponential) and more generally in any unital
Banach algebra . 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
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 and its associated
Lie algebra , the
exponential map is a map
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 supplies the necessary correction terms.
Transcendency
The function is not in (that is, is not the quotient of two polynomials with complex coefficients).
If are distinct complex numbers, then are linearly independent over . It follows that 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
with
floating-point arithmetic may lead to the loss of (possibly all)
significant figures, producing a large calculation error, possibly even a meaningless result.
Following a proposal by
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
one may use the Taylor series of
:
This was first implemented in 1979 in the
Hewlett-Packard HP-41C calculator, and provided by several calculators,
operating system
An operating system (OS) is system software that manages computer hardware, software resources, and provides common daemon (computing), services for computer programs.
Time-sharing operating systems scheduler (computing), schedule tasks for ef ...
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:
and
.
A similar approach has been used for the logarithm (see
lnp1).
An identity in terms of the
hyperbolic tangent
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
,
gives a high-precision value for small values of on systems that do not implement .
Alternatively, this expression can be used:
:
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
*
-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)