Exponential function

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The natural exponential function y = e Exp.svg
The natural exponential function y = e
Exponential functions with bases 2 and 1/2 Exponenciala priklad.png
Exponential functions with bases 2 and 1/2

In mathematics, the exponential function is the function where e = 2.71828... is Euler's number.

Contents

More generally, an exponential function is a function of the form

where b is a positive real number, and the argument x occurs as an exponent. For real numbers c and d, a function of the form is also an exponential function, since it can be rewritten as

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

As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:

For b > 1, the function is increasing (as depicted for b = e and b = 2), because makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = 1/2); and for b = 1 the function is constant.

The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function is its own derivative:

This function, also denoted as exp x, is called the "natural exponential function", [1] [2] [3] 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. The graph of is upward-sloping, and increases faster as x increases. [4] The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point. Its inverse function is the natural logarithm, denoted [nb 1] [nb 2] or because of this, some old texts [5] refer to the exponential function as the antilogarithm.

The exponential function satisfies the fundamental multiplicative identity (which can be extended to complex-valued exponents as well):

It can be shown that every continuous, nonzero solution of the functional equation is an exponential function, with The multiplicative identity, along with the definition , shows that for positive integers n, relating the exponential function to the elementary notion of exponentiation.

The argument of the exponential function can be any real or complex number, or even an entirely different kind of mathematical object (for example, a square matrix).

The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". [6] 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, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

Formal definition

The exponential function (in blue), and the sum of the first n + 1 terms of its power series (in red). Exp series.gif
The exponential function (in blue), and the sum of the first n + 1 terms of its power series (in red).

The real exponential function can be characterized in a variety of equivalent ways. It is commonly defined by the following power series: [6] [7]

Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers z ∈ ℂ (see § Complex plane for the extension of to the complex plane). The constant e can then be defined as

The term-by-term differentiation of this power series reveals that for all real x, leading to another common characterization of as the unique solution of the differential equation

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: [8] [7]

Overview

The red curve is the exponential function. The black horizontal lines show where it crosses the green vertical lines. Animation of exponential function.gif
The red curve is the exponential function. The black horizontal lines show where it crosses the green vertical lines.

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 [9] to the number

now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function. [9]

If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is x/12 times the current value, so each month the total value is multiplied by (1 + x/12), and the value at the end of the year is (1 + x/12)12. If instead interest is compounded daily, this becomes (1 + x/365)365. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function,

first given by Leonhard Euler. [8] This is one of a number of characterizations of the exponential function; others involve series or differential equations.

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity,

which justifies the notation ex for exp x.

The 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 or exponential decay.

The exponential function extends to an entire function on the complex plane. Euler's formula relates its values at purely imaginary arguments to trigonometric functions. The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.

Derivatives and differential equations

The derivative of the exponential function is equal to the value of the function. From any point P on the curve (blue), let a tangent line (red), and a vertical line (green) with height h be drawn, forming a right triangle with a base b on the x-axis. Since the slope of the red tangent line (the derivative) at P is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, h must be equal to the ratio of h to b. Therefore, the base b must always be 1. Exp tangent.svg
The derivative of the exponential function is equal to the value of the function. From any point P on the curve (blue), let a tangent line (red), and a vertical line (green) with height h be drawn, forming a right triangle with a base b on the x-axis. Since the slope of the red tangent line (the derivative) at P is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, h must be equal to the ratio of h to b. Therefore, the base b must always be 1.

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 x = 0. That is,

Functions of the form cex for constant c are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Other ways of saying the same thing include:

If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant k, a function f: RR satisfies f′ = kf if and only if f(x) = cekx for some constant c. The constant k is called the decay constant, disintegration constant, [10] rate constant, [11] or transformation constant. [12]

Furthermore, for any differentiable function f(x), we find, by the chain rule:

Continued fractions for ex

A continued fraction for ex can be obtained via an identity of Euler:

The following generalized continued fraction for ez converges more quickly: [13]

or, by applying the substitution z = x/y:

with a special case for z = 2:

This formula also converges, though more slowly, for z > 2. For example:

Complex plane

A complex plot of
z
-
exp
[?]
z
{\displaystyle z\mapsto \exp z}
, with the phase
Arg
[?]
exp
[?]
z
{\displaystyle \operatorname {Arg} \exp z}
represented by varying hue. The transition from dark to light colors shows that
|
exp
[?]
z
|
{\displaystyle \left|\exp z\right|}
is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that
z
-
exp
[?]
z
{\displaystyle z\mapsto \exp z}
is periodic in the imaginary part of
z
{\displaystyle z}
. Complex exp.jpg
A complex plot of , with the phase represented by varying hue. The transition from dark to light colors shows that is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that is periodic in the imaginary part of .

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 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 to complex arguments.

In particular, when z = it (t 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 cos t and sin t, respectively.

This correspondence provides motivation for defining cosine and sine for all complex arguments in terms of and the equivalent power series: [14]

The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test 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, 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:

.

We could alternatively define the complex exponential function based on this relationship. If z = x + iy, where x and y are both real, then we could define its exponential as

where exp, cos, and sin on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means. [15]

For , the relationship holds, so that for real and maps the real line (mod 2π) 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 z = 1 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 2πi and holds for all .

When its domain is extended from the real line to the complex plane, the exponential function retains the following properties:

.

Extending the natural logarithm to complex arguments yields the complex logarithm log z, which is a multivalued function.

We can then define a more general exponentiation:

for all complex numbers z and w. This is also a multivalued function, even when z is real. This distinction is problematic, as the multivalued functions log z and zw are easily confused with their single-valued equivalents when substituting a real number for z. The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:

(ez)w
ezw
, but rather (ez)w
= e(z + 2niπ)w
multivalued over integers n

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. 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.

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.

The second image shows how the domain complex plane is mapped into the range complex plane:

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 ±2π, this image also better depicts the 2π periodicity in the imaginary value.

Computation of ab where both a and b are complex

Complex exponentiation ab can be defined by converting a to polar coordinates and using the identity (eln a)b
= ab
:

However, when b is not an integer, this function is multivalued, because θ 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 B. In this setting, e0 = 1, and ex is invertible with inverse ex for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y.

Some alternative definitions lead to the same function. For instance, ex can be defined as

Or ex can be defined as fx(1), where fx : RB is the solution to the differential equation dfx/dt(t) = xfx(t), with initial condition fx(0) = 1; it follows that fx(t) = etx for every t in R.

Lie algebras

Given a Lie group G and its associated Lie algebra , the exponential map is a map G satisfying similar properties. In fact, since R 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 GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

The identity exp(x + y) = exp x exp y can fail for Lie algebra elements x and y that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.

Transcendency

The function ez is not in C(z) (that is, is not the quotient of two polynomials with complex coefficients).

For n distinct complex numbers {a1, …, an}, the set {ea1z, …, eanz} is linearly independent over C(z).

The function ez is transcendental over C(z).

Computation

When computing (an approximation of) the exponential function near the argument 0, 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 ex − 1 directly, bypassing computation of ex. 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, [16] [17] operating systems (for example Berkeley UNIX 4.3BSD [18] ), computer algebra systems, and programming languages (for example C99). [19]

In addition to base e, 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). [nb 3]

An identity in terms of the hyperbolic tangent,

gives a high-precision value for small values of x on systems that do not implement expm1(x).

See also

Notes

  1. In pure mathematics, the notation log x generally refers to the natural logarithm of x or a logarithm in general if the base is immaterial.
  2. The notation ln x is the ISO standard and is prevalent in the natural sciences and secondary education (US). However, some mathematicians (for example, Paul Halmos) have criticized this notation and prefer to use log x for the natural logarithm of x.
  3. A similar approach to reduce round-off errors of calculations for certain input values of trigonometric functions consists of using the less common trigonometric functions versine, vercosine, coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant.

Related Research Articles

Complex number Element of a number system in which –1 has a square root

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Eulers formula Expression of the complex exponential in terms of sine and cosine

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 any real number x:

<span class="texhtml mvar" style="font-style:italic;">e</span> (mathematical constant) e ≈ 2.71828..., base of the natural logarithm

The number e, 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 the natural logarithm. It is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

Gamma function Extension of the factorial function

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n,

Logarithm Inverse of the exponential function, which maps products to sums

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10(1000) = 3. The logarithm of x to baseb is denoted as logb(x), or without parentheses, logbx, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

Natural logarithm Logarithm to the base of the mathematical constant e

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

Lambert <i>W</i> function Multivalued "function"

In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the inverse relation of the function f(w) = wew, where w is any complex number and ew is the exponential function.

Exponentiation Mathematical operation

Exponentiation is a mathematical operation, written as bn, involving two numbers, the baseb and the exponent or powern, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:

In calculus, the power rule is used to differentiate functions of the form , whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives.

In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation

In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula

Tetration Repeated or iterated exponentiation

In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

Exponential integral Special function defined by an integral

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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 of and equivalent to each other. As a special case of these considerations, it will be demonstrated that the three most common definitions given for the mathematical constant e are equivalent to each other.

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.

Exponential type type of complex function with growth bounded by an exponential function

In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function eC|z| for some real-valued constant C as |z| → ∞. When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of Ψ-type for a general function Ψ(z) as opposed to ez.

Complex logarithm Logarithm of a complex number

In the branch of mathematics known as complex analysis, a complex logarithm is an analogue for nonzero complex numbers of the logarithm of a positive real number. The term refers to one of the following:

In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region if and only if is analytic on and has a harmonic majorant on where . Being the ratio of two bounded analytic functions is a sufficient condition for a function to be of bounded type, and if is simply connected the condition is also necessary.

Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.

In mathematics, the field of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each log-exp transseries represents a formal asymptotic behavior, and it can be manipulated formally, and when it converges, corresponds to actual behavior. Transseries can also be convenient for representing functions. Through their inclusion of exponentiation and logarithms, transseries are a strong generalization of the power series at infinity and other similar asymptotic expansions.

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