The exponential function is a mathematical
function denoted by
or
(where the argument is written as an
exponent
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 re ...
). 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 re ...
(repeated multiplication), but modern definitions (there are
several equivalent characterizations) allow it to be rigorously extended to all real arguments, including
irrational numbers
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
. 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 mathemat ...
led mathematician
Walter Rudin to opine that the exponential function is "the most important function in mathematics".
[
The exponential function satisfies the exponentiation identity
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
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...
called Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
.
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
. 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 ...
, 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 for ...
.
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
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are ofte ...
or a Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
).
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 using ...
, 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 disti ...
, 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 rel ...
, 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 practical disciplines (includin ...
, chemistry, engineering
Engineering is the use of scientific method, 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 rang ...
, mathematical biology
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...
, 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 analy ...
.
The real exponential function is a bijection from to . Its inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
is the natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, denoted [ ][ or because of this, some old texts][ refer to the exponential function as the antilogarithm.
]
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 discre ...
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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
. 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 used ...
of the tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
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
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
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, a ...
satisfying the initial condition
Based on this characterization, the chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
shows that its inverse function, the natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, 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
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
and the power series definition, the exponential function can also be defined as the following limit:[
It can be shown that every continuous, 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
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 compare ...
to its current value. One such situation is continuously compounded interest, 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 in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the L ...
in 1683[ to the number
now known as . Later, in 1697, ]Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating ...
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 in ma ...
.[
This is one of a number of ]characterizations of the exponential function In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent ...
; 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, a ...
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 re ...
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
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
or exponential decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
.
The exponential function extends to an entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
on the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
. 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 for ...
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 ...
. 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
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
or a Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
.
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 C ...
). 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, a ...
.
* 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 s ...
.
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 compare ...
to its size—as is the case in unlimited population growth (see Malthusian catastrophe), 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 disti ...
, 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 consid ...
—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
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
:
Continued fractions for
A continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integ ...
for can be obtained via an identity of Euler:
The following generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.
A g ...
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 mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
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
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 ...
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 ...
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 function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
s (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, 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 for ...
:
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 mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
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
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to b ...
, which is a multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
.
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
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
in the complex plane to a logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
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), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
. 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
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential giv ...
) and more generally in any unital Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
. 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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
and its associated Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
, 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
Transcendence, transcendent, or transcendental may refer to:
Mathematics
* Transcendental number, a number that is not the root of any polynomial with rational coefficients
* Algebraic element or transcendental element, an element of a field exten ...
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
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
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 digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expre ...
, 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 mathematician and computer scientist, who received the Turing Award in 1989 for "''his fundamental contributions to numerical analysis''",
was named an ACM Fellow in 1994, and induc ...
, 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, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor se ...
one may use the Taylor series of :
This was first implemented in 1979 in the Hewlett-Packard HP-41C
The HP-41C series are programmable, expandable, continuous memory handheld RPN calculators made by Hewlett-Packard from 1979 to 1990. The original model, HP-41C, was the first of its kind to offer alphanumeric display capabilities. Later cam ...
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 system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...
s, and programming languages (for example C99).[
In addition to base , the ]IEEE 754-2008
The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operati ...
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
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It i ...
* Half-exponential function, a compositional square root of an exponential function
* List of exponential topics {{Short description, none
This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics.
* Accelerating change
* Approximating natural exponents (log base e)
* Artin–Hasse exponential
* Bacterial growth
* Bak ...
* 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)