# Exponential function

In mathematics, an **exponential function** is a function of the form

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

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 real exponential function is a constant and the derivative is zero because for positive a and *b* > 1 the real exponential functions are monotonically increasing (as depicted for *b* = *e* and *b* = 2), because the derivative is greater than zero for all arguments, and for *b* < 1 they are monotonically decreasing (as depicted for *b* = 1/2), because the derivative is less than zero for all arguments.

The constant *e* = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself:

Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function",[1][2] or simply, "the exponential function" and denoted by

While both notations are common, the former notation is generally used for simpler exponents, while the latter tends to be used when the exponent is a complicated expression.

The exponential function satisfies the fundamental multiplicative identity

This identity extends to complex-valued exponents. It can be shown that every continuous, nonzero solution of the functional equation is an exponential function, with The fundamental multiplicative identity, along with the definition of the number e as *e*^{1}, shows that for positive integers n and relates 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 matrix).

Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics".[3] 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; thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.

The graph of is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can be arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y-coordinate at that point, as implied by its derivative function (*see above*). Its inverse function is the natural logarithm, denoted [nb 1] [nb 2] or because of this, some old texts[4] refer to the exponential function as the **antilogarithm**.

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## Formal definition

The real exponential function can be characterized in a variety of equivalent ways. Most commonly, it is defined by the following power series:[3]

Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers (see below 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:[5]

## 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[6] to the number

now known as *e*. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.[6]

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.[5] 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 *e*^{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 importance of the exponential function in mathematics and the sciences stems mainly from its definition as the unique function which is equal to its derivative and is equal to 1 when *x* = 0. That is,

Functions of the form *ce*^{x} 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:

- 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
*x*is equal to the value of the function at*x*. - The function solves the differential equation
*y*′ =*y*. - exp is a fixed point of derivative as a functional.

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*: **R** → **R** satisfies *f*′ = *kf* if and only if *f*(*x*) = *ce*^{kx} for some constant *c*.

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

## Continued fractions for *e*^{x}

*e*

^{x}

A continued fraction for *e*^{x} can be obtained via an identity of Euler:

The following generalized continued fraction for *e*^{z} converges more quickly:[7]

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

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:

Term-wise multiplication of two copies of these 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:

- for all

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

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 *defining* cosine and sine for all complex arguments in terms of and the equivalent power series:[8]

- and for all

The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test and are therefore entire functions (*i.e.*, 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:

- for all

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 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.[9]

For , the relationship holds, so that for real and maps the real line (mod ) to the unit circle. Based on the relationship between and the unit circle, it is easy to see that, restricted to real arguments, the definitions of sine and cosine given above coincide with their more elementary definitions based on 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 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 *z*^{w} 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:

- (
*e*)^{z}^{w}_{}≠*e*, but rather (^{zw}*e*)^{z}^{w}_{}=*e*^{(z + 2πin)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.

- Checker board key:

- Projection into the , , and dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).
- Projection into the , , and dimensions, producing a spiral shape. ( range extended to ±2π, 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 ±2π, this image also better depicts the 2π periodicity in the imaginary value.

### Computation of *a*^{b} where both *a* and *b* are complex

*a*

^{b}where both

*a*and

*b*are complex

Complex exponentiation *a*^{b} can be defined by converting *a* to polar coordinates and using the identity (*e*^{ln a})^{b}_{} = *a*^{b}:

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, *e*^{0} = 1, and *e*^{x} is invertible with inverse *e*^{−x} for any *x* in *B*. If *xy* = *yx*, then *e*^{x + y} = *e*^{x}*e*^{y}, but this identity can fail for noncommuting *x* and *y*.

Some alternative definitions lead to the same function. For instance, *e*^{x} can be defined as

Or *e*^{x} can be defined as *f*_{x}(1), where *f*_{x}: **R**→*B* is the solution to the differential equation *df _{x}*/

*dt*(

*t*) =

*x f*

_{x}(

*t*), with initial condition

*f*

_{x}(0) = 1; it follows that

*f*

_{x}(

*t*) =

*e*

^{tx}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 *e*^{z} is not in **C**(*z*) (i.e., is not the quotient of two polynomials with complex coefficients).

For *n* distinct complex numbers {*a*_{1}, …, *a*_{n}}, the set {*e*^{a1z}, …, *e*^{anz}} is linearly independent over **C**(*z*).

The function *e*^{z} 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 *e ^{x}* − 1 directly, bypassing computation of

*e*

^{x}. 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,[10][11] operating systems (for example Berkeley UNIX 4.3BSD[12]), computer algebra systems, and programming languages (for example C99).[13]

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

- Approximating natural exponents (log base
*e*) - Carlitz exponential, a characteristic
*p*analogue - Double exponential function
- Exponential field
- Half-exponential function, a compositional square root of an exponential function
- Linear function
- List of exponential topics
- List of integrals of exponential functions
- Mittag-Leffler function, a generalization of the exponential function
- Multiplicative calculus
*p*-adic exponential function- Polynomial function
- Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions
- Tetration
- Unrestricted algorithm for the computation of the exponential function

## Notes

- 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. - The notation ln
*x*is the ISO standard and is prevalent in the natural sciences and secondary education (US). However, some mathematicians (e.g., Paul Halmos) have criticized this notation and prefer to use log*x*for the natural logarithm of x. - 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.

## References

- Goldstein, Larry Joel; Lay, David C.; Schneider, David I.; Asmar, Nakhle H. (2006).
*Brief calculus and its applications*(11th ed.). Prentice–Hall. ISBN 978-0-13-191965-5. (467 pages) - Courant; Robbins (1996). Stewart (ed.).
*What is Mathematics? An Elementary Approach to Ideas and Methods*(2nd revised ed.). Oxford University Press. p. 448. ISBN 978-0-13-191965-5.*This natural exponential function is identical with its derivative.*This is really the source of all the properties of the exponential function, and the basic reason for its importance in applications… - Rudin, Walter (1987).
*Real and complex analysis*(3rd ed.). New York: McGraw-Hill. p. 1. ISBN 978-0-07-054234-1. - Converse, Henry Augustus; Durell, Fletcher (1911).
*Plane and Spherical Trigonometry*. Durell's mathematical series. C. E. Merrill Company. p. 12.Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) ...

- Maor, Eli.
*e: the Story of a Number*. p. 156. - O'Connor, John J.; Robertson, Edmund F. (September 2001). "The number e".
*School of Mathematics and Statistics*. University of St Andrews, Scotland. Retrieved 2011-06-13. - Lorentzen, L.; Waadeland, H. (2008). "A.2.2 The exponential function.".
*Continued Fractions*. Atlantis Studies in Mathematics. p. 268. doi:10.2991/978-94-91216-37-4. - Rudin, Walter (1976).
*Principles of Mathematical Analysis*. New York: McGraw-Hill. p. 182. ISBN 978-0-07054235-8. - Apostol, Tom M. (1974).
*Mathematical Analysis*(2nd ed.). Reading, Mass.: Addison Wesley. p. 19. ISBN 978-0-20100288-1. -
*HP 48G Series – Advanced User's Reference Manual (AUR)*(4 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2. Retrieved 2015-09-06. -
*HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR)*(2 ed.). Hewlett-Packard. 2009-07-14 [2005]. HP F2228-90010. Retrieved 2015-10-10. - Beebe, Nelson H. F. (2017-08-22). "Chapter 10.2. Exponential near zero".
*The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library*(1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. pp. 273–282. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446.Berkeley UNIX 4.3BSD introduced the expm1() function in 1987.

- Beebe, Nelson H. F. (2002-07-09). "Computation of expm1 = exp(x)−1" (PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.

## External links

- Hazewinkel, Michiel, ed. (2001) [1994], "Exponential function",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - "Complex exponential function".
*PlanetMath*. - "Derivative of exponential function".
*PlanetMath*. - Weisstein, Eric W. "Exponential Function".
*MathWorld*.