Exponential function explained

In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. The exponential of a variable is denoted or, with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant number , the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: it maps the additive identity to the multiplicative identity, and the exponential of a sum is equal to the product of separate exponentials, . Its inverse function, the natural logarithm, or, converts products to sums: .

Other functions of the general form, with base, are also commonly called exponential functions, and share the property of converting addition to multiplication, . Where these two meanings might be confused, the exponential function of base is occasionally called the natural exponential function, matching the name natural logarithm. The generalization of the standard exponent notation to arbitrary real numbers as exponents, is usually formally defined in terms of the exponential and natural logarithm functions, as . The "natural" base is the unique base satisfying the criterion that the exponential function's derivative equals its value,, which simplifies definitions and eliminates extraneous constants when using exponential functions in calculus.

Quantities which change over time in proportion to their value, for example the balance of a bank account bearing compound interest, the size of a bacterial population, the temperature of an object relative to its environment, or the amount of a radioactive substance, can be modeled using functions of the form, also sometimes called exponential functions; these quantities undergo exponential growth if is positive or exponential decay if is negative.

The exponential function can be generalized to accept a complex number as its argument. This reveals a relation between the multiplication of complex numbers and rotation in the Euclidean plane, Euler's formula : the exponential of an imaginary number is a point on the complex unit circle at angle from the real axis. The identities of trigonometry can thus be translated into identities involving exponentials of imaginary quantities. The complex function is a conformal map from an infinite strip of the complex plane (which periodically repeats in the imaginary direction) onto the whole complex plane except for .

The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras. Some old texts refer to the exponential function as the antilogarithm.

Graph

The graph of

is upward-sloping, and increases faster as increases.^[1] 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 of the tangent to the graph at each point is equal to its height (its -coordinate) at that point.

Definitions and fundamental properties

See also: Characterizations of the exponential function. There are several different definitions of the exponential function, which are all equivalent, although of very different nature.

One of the simplest definitions is: The exponential function is the unique differentiable function that equals its derivative, and takes the value for the value of its variable.

This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.

Uniqueness: If and are two functions satisfying the above definition, then the derivative of is zero everywhere by the quotient rule. It follows that is constant, and this constant is since .

The exponential function is the inverse function of the natural logarithm. The inverse function theorem implies that the natural logarithm has an inverse function, that satisfies the above definition. This is a first proof of existence. Therefore, one has

for every real number and evey positive real number

The exponential function is the sum of a power series:^[2] where

is the factorial of (the product of the first positive integers). This series is absolutely convergent for every per the ratio test. So, the derivative of the sum can be computed by term-by-term derivation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that the exponential function is defined for every, and is everywhere the sum of its Maclaurin series.

The exponential satisfies the functional equation:$$\backslash exp(x+y)=\; \backslash exp(x)\backslash cdot\; \backslash exp(y).$$This results from the uniqueness and the fact that the function

satisfies the above definition. It can be proved that a function that satisfies this functional equation is the exponential function if its derivative at is and the function is either continuous or monotonic

Positiveness: For every, one has, since the functional equation implies . It results that the exponential function is positive (since, if one would have for some, the intermediate value theorem would imply the existence of some such that . It results also that the exponential function is monotonically increasing.

Extension of exponentiation to positive real bases: Let be a positive real number. The exponential function and the naturel logarithm being the inverse each of the other, one has

If is an integer, the functional equation of the logarithm implies $$b^n=\backslash exp(\backslash ln\; b^n)=\; \backslash exp(n\backslash ln\; b).$$Since the right-most expression is defined if is any real number, this allows defining for every positive real number and every real number :$$b^x=\backslash exp(x\backslash ln\; b).$$In particular, if is the Euler's number one has (inverse function) and thus $$e^x=\backslash exp(x).$$ This shows the equivalence of the two notations for the exponential function.

The exponential function is the limit$$\backslash exp(x)=\backslash lim\_\; \backslash left(1+\backslash frac\; xn\backslash right)^n,$$where

takes only integer values (otherwise, the exponentiation would require the exponential function to be defined). By continuity of the logarithm, this can be proved by taking logarithms and proving $$x=\backslash lim\_\backslash ln\; \backslash left(1+\backslash frac\; xn\backslash right)^n=\; \backslash lim\_n\backslash ln\; \backslash left(1+\backslash frac\; xn\backslash right),$$for example with Taylor's theorem.

Generalized exponential functions

The exponential function

is sometimes called the natural exponential function to distinguish it from generalizations that are also often called exponential functions. These functions have generally the form$$f(x)=ab^,$$where is a positive real number.

Such a function can be expressed in terms of the natural exponential function as $$f(x)=Ce^,$$by setting$C=ae^$ and $\backslash alpha=c\backslash ln(b).$ In applications in empirical sciences, it is common the use the base instead of, which amounts to replace with and with in what precedes.

A property of these generalized exponential function is that their derivative is directly proportional to the function; that is,$$\backslash frac=\backslash alpha,$$with above notation. This property characterize the generalized exponential functions.

Also, if the variable is increased by a constant value, the value of the funtion is multiplied by a constant value depending on :$$Ce^=e^\backslash cdot\; Ce^.$$

In many applications, this multiplivative factor is measured by the value of for which it is of ; that is, $$h=\backslash frac$$

.This is in particular the case of the half time in physics.

When the argument and the value of an exponential function represent physical quantities, the function is changed by a change of measurement units. Conversely, given two exponential functions that have the same sign before the variable in the exponent, one can transform the first one into the second one by changing of measurement units. This means that for modeling physical processes, there is essentially two exponential functions, one for exponential decay and one for exponential growth.

For example, exponential decay is commonly modeled by the function$$f(t)=f(t\_0)2^,$$where is the half time. If one takes as the time origin, as time unit, and as unit for the measured quantity, the function becomes $$f\_2(t)=\backslash frac\; 1.$$For every positive base, another change of units transform the function into the function $$f\_b(t)=\backslash frac\; 1.$$Similarly, with a convenient change of units, exponential growth can always be modeled as the function, or, for every positive base, as the function $$g\_b(t)=b^t.$$

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$$\backslash lim\_\backslash left(1\; +\; \backslash frac\backslash right)^$$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 definition of the exponential function,$$\backslash exp\; x\; =\; \backslash lim\_\backslash left(1\; +\; \backslash frac\backslash right)^$$first given by Leonhard Euler.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 is the reciprocal of . For example, from the differential equation definition, when and its derivative using the product rule is for all, so for all .

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity. For example, from the power series definition, expanded by the Binomial theorem,$$\backslash exp(x\; +\; y)=\; \backslash sum\_^\; \backslash frac=\; \backslash sum\_^\; \backslash sum\_^n\backslash frac\; \backslash frac=\; \backslash sum\_^\; \backslash sum\_^\; \backslash frac=\; \backslash exp\; x\; \backslash cdot\; \backslash exp\; y\backslash ,.$$This justifies the exponential notation for .

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 is expressible in terms of the exponential function. This derivative 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 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,$$\backslash frace^x\; =\; e^x\; \backslash quad\backslash text\backslash quad\; e^0=1.$$

Functions of the form for constant are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). 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 .
• is a fixed point of derivative as a linear operator on function space.

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.

More generally, 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:$$\backslash frac\; e^\; =\; f\text{'}(x)\backslash ,e^.$$

Continued fractions for

A continued fraction for can be obtained via an identity of Euler:$$e^x\; =\; 1\; +\; \backslash cfrac$$

The following generalized continued fraction for converges more quickly:$$e^z\; =\; 1\; +\; \backslash cfrac$$

or, by applying the substitution :$$e^\backslash frac\; =\; 1\; +\; \backslash cfrac$$with a special case for :$$e^2\; =\; 1\; +\; \backslash cfrac\; =\; 7\; +\; \backslash cfrac$$

This formula also converges, though more slowly, for . For example:$$e^3\; =\; 1\; +\; \backslash cfrac\; =\; 13\; +\; \backslash cfrac$$

Complex exponential

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:$$\backslash exp\; z\; :=\; \backslash sum\_^\backslash infty\backslash frac$$

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:$$\backslash exp\; z\; :=\; \backslash lim\_\backslash left(1+\backslash frac\backslash right)^n$$

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:$$\backslash exp(w+z)=\backslash exp\; w\backslash exp\; z\; \backslash text\; w,z\backslash in\backslash mathbb$$

The definition of the complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments.

In particular, when (real), the series definition yields the expansion$$\backslash exp(it)\; =\; \backslash left(1-\backslash frac+\backslash frac-\backslash frac+\backslash cdots\; \backslash right)\; +\; i\backslash left(t\; -\; \backslash frac\; +\; \backslash frac\; -\; \backslash frac+\backslash cdots\backslash right).$$

In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of and, respectively.

This correspondence provides motivation for cosine and sine for all complex arguments in terms of

and the equivalent power series:

for all $z\backslash in\backslash mathbb.$

The functions,, and 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:$$\backslash exp(iz)=\backslash cos\; z+i\backslash sin\; z\; \backslash text\; z\backslash in\backslash mathbb.$$

We could alternatively define the complex exponential function based on this relationship. If, where and are both real, then we could define its exponential as$$\backslash exp\; z\; =\; \backslash exp(x+iy)\; :=\; (\backslash exp\; x)(\backslash cos\; y\; +\; i\; \backslash sin\; y)$$where,, and on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.

For

, 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$$\backslash int\_0^|\backslash gamma\text{'}(t)|\; \backslash ,\; dt\; =\; \backslash int\_0^\; |i\backslash exp(it)|\; \backslash ,\; dt\; =\; t\_0,$$starting from in the complex plane and going counterclockwise. Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions.

The complex exponential function is periodic with period and

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 $w,z\backslash in\backslash mathbb\; C.$

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

We can then define a more general exponentiation:$$z^w\; =\; e^$$for all complex numbers and . This is also a multivalued function, even when is real. This distinction is problematic, as the multivalued functions and are easily confused with their single-valued equivalents when substituting a real number for . The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:

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:$$v\; +\; i\; w\; =\; \backslash exp(x\; +\; i\; y)$$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:

• 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.

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$$\backslash lim\_\; \backslash left(1\; +\; \backslash frac\; \backslash right)^n\; .$$

Or can be defined as, where is the solution to the differential equation, with initial condition ; it follows that for every in .

Lie algebras

, 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 the rational function ring

: it is not the quotient of two polynomials with complex coefficients.

If are distinct complex numbers, then are linearly independent over

, and hence is transcendental over .

Computation

The Taylor series definition above is generally efficient for computing (an approximation of)

. However, when computing 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 relative 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, which computes directly, bypassing computation of . For example,one may use the Taylor series:$$e^x-1=x+\backslash frac\; 2\; +\; \backslash frac6+\backslash cdots\; +\backslash frac+\backslash cdots.$$

This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators, operating systems (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,$$\backslash operatorname\; (x)\; =\; e^x\; -\; 1\; =\; \backslash frac,$$gives a high-precision value for small values of on systems that do not implement .

See also

• Carlitz exponential, a characteristic analogue
  • • Gaussian function
• Half-exponential function, a compositional square root of an exponential function
• - Used for solving exponential equations
• 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
• Phase factor

Notes and References

1. Web site: Exponential Function Reference. 2020-08-28. www.mathsisfun.com.
2. Web site: Weisstein. Eric W.. Exponential Function. 2020-08-28. mathworld.wolfram.com. en.