Complex Exponential

The exponential function is a mathematical function denoted by $f(x)=\exp(x)$ or $e^x$ (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".

The exponential function satisfies the exponentiation identity
$e^ = e^x e^y \text x,y\in\mathbb,$
which, along with the definition $e = \exp(1)$, shows that $e^n=\underbrace_$ for positive integers , and relates the exponential function to the elementary notion of exponentiation. The base of the exponential function, its value at 1, $e = \exp(1)$, is a ubiquitous mathematical constant called Euler's number.

While other continuous nonzero functions $f:\mathbb\to\mathbb$ 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, $\exp'(x)=\exp(x)$ for all real , and $\exp(0)=1.$ Thus, exp is sometimes called the natural exponential function to distinguish it from these other exponential functions, which are the functions of the form $f(x) = ab^x,$ where the base is a positive real number. The relation $b^x = e^$ 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 $\exp:\mathbb\to\mathbb$ to be defined. The complex exponential function takes on all complex values except for 0 and is closely related to the complex trigonometric functions, as shown by Euler's formula.

Motivated by more abstract properties and characterizations of the exponential function, the exponential can be generalized to and defined for entirely different kinds of mathematical objects (for example, a square matrix or a Lie algebra).

In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, 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, computer science, chemistry, engineering, mathematical biology, and economics.

The real exponential function is a bijection from $\mathbb$ to $(0;\infty)$. Its inverse function is the natural logarithm, denoted $\ln,$ $\log,$ or $\log_e;$ because of this, some old texts refer to the exponential function as the antilogarithm.

Graph

The graph of $y=e^x$ is upward-sloping, and increases faster as increases. The graph always lies above the -axis, but becomes arbitrarily close to it for large negative ; thus, the -axis is a horizontal asymptote. The equation $\tfrace^x = e^x$ means that the slope of the tangent to the graph at each point is equal to its -coordinate at that point.

Relation to more general exponential functions

The exponential function $f(x) = e^x$ 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 ,
$ab^x := ae^$

As functions of a real variable, exponential functions are uniquely characterized by the fact that the derivative of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base :
$\frac b^x \ \ \frac e^ = e^ \ln (b) = b^x \ln (b).$

For , the function $b^x$ is increasing (as depicted for and ), because $\ln b>0$ makes the derivative always positive; while for , the function is decreasing (as depicted for ); and for the function is constant.

Euler's number is the unique base for which the constant of proportionality is 1, since $\ln(e) = 1$, so that the function is its own derivative:
$\frac e^x = e^x \ln (e) = e^x.$

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 $b^x = e^$, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by
$x\mapsto e^x$ or $x\mapsto \exp x.$

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 $f(x) = a b^$ is also an exponential function, since it can be rewritten as
$a b^ = \left(a b^d\right) \left(b^c\right)^x.$

Formal definition

The real exponential function $\exp\colon\mathbb\to\mathbb$ can be characterized in a variety of equivalent ways. It is commonly defined by the following power series:
$\exp x := \sum_^ \frac = 1 + x + \frac + \frac + \frac + \cdots$

Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers $z\in\mathbb$ (see for the extension of $\exp x$ to the complex plane). The constant can then be defined as $e = \exp 1 = \sum_^\infty(1/k!).$

The term-by-term differentiation of this power series reveals that $\frac\exp x = \exp x$ for all real , leading to another common characterization of $\exp x$ as the unique solution of the differential equation
$y'(x) = y(x),$
satisfying the initial condition $y(0) = 1.$

Based on this characterization, the chain rule shows that its inverse function, the natural logarithm, satisfies $\frac\log_e y = 1/y$ for $y > 0,$ or $\log_e y = \int_1^y \frac\,.$ This relationship leads to a less common definition of the real exponential function $\exp x$ as the solution $y$ to the equation
$x = \int_1^y \frac \, dt.$

By way of the binomial theorem and the power series definition, the exponential function can also be defined as the following limit:
$\exp x = \lim_ \left(1 + \frac\right)^n.$

It can be shown that every continuous, nonzero solution of the functional equation $f(x+y)=f(x)f(y)$ is an exponential function, $f: \R \to \R,\ x \mapsto e^,$ with $k\in\mathbb.$

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
$\lim_\left(1 + \frac\right)^$
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,
$\exp x = \lim_\left(1 + \frac\right)^$
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 the exponential function obeys the basic exponentiation identity,
$\exp(x + y) = \exp x \cdot \exp y$
which justifies the 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 (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 property as the unique function which is equal to its derivative and is equal to 1 when . That is,
$\frace^x = e^x \quad\text\quad e^0=1.$

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 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 , 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:
$\frac e^ = f'(x)e^.$

Continued fractions for

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

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

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

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

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:
$\exp z := \sum_^\infty\frac$

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:
$\exp z := \lim_\left(1+\frac\right)^n$

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

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
$\exp(it) = \left( 1-\frac+\frac-\frac+\cdots \right) + i\left(t - \frac + \frac - \frac+\cdots\right).$

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 $\exp(\pm iz)$ and the equivalent power series:
\begin
& \cos z:= \frac = \sum_^\infty (-1)^k \frac, \\ pt \text \quad & \sin z := \frac =\sum_^\infty (-1)^k\frac
\end

for all $z\in\mathbb.$

The functions , , and so defined have infinite radii of convergence by the ratio test and are therefore entire functions (that is, holomorphic on $\mathbb$). The range of the exponential function is $\mathbb\setminus \$, while the ranges of the complex sine and cosine functions are both $\mathbb$ in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of $\mathbb$, or $\mathbb$ excluding one lacunary value.

These definitions for the exponential and trigonometric functions lead trivially to Euler's formula:
$\exp(iz)=\cos z+i\sin z \text z\in\mathbb.$

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
$\exp z = \exp(x+iy) := (\exp x)(\cos y + i \sin y)$
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 $t\in\R$, the relationship $\overline=\exp(-it)$ holds, so that $\left, \exp(it)\ = 1$ for real $t$ and $t \mapsto \exp(it)$ maps the real line (mod ) to the unit circle in the complex plane. Moreover, going from $t = 0$ to $t = t_0$, the curve defined by $\gamma(t)=\exp(it)$ traces a segment of the unit circle of length
$\int_0^, \gamma'(t), \, dt = \int_0^ , i\exp(it), \, dt = t_0,$
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 $\exp(z+2\pi i k)=\exp z$ holds for all $z \in \mathbb, k \in \mathbb$.

When its domain is extended from the real line to the complex plane, the exponential function retains the following properties:
\begin
& e^ = e^z e^w\, \\ pt & e^0 = 1\, \\ pt & e^z \ne 0 \\ pt & \frac e^z = e^z \\ pt & \left(e^z\right)^n = e^, n \in \mathbb
\end

for all $w,z\in\mathbb C.$

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.

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:
$v + i w = \exp(x + i y)$
the graph of the exponential function is a two-dimensional surface curving through four dimensions.

Starting with a color-coded portion of the $xy$ 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:
$x> 0:\; \text$
$x< 0:\; \text$
$y> 0:\; \text$
$y< 0:\; \text$
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 $x$, $v$, and $w$ 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 $y$, $v$, and $w$ dimensions, producing a spiral shape. ($y$ 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 $x$ axis is mapped to the positive real $v$ axis
* the imaginary $y$ 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 $x$ axis. It shows the graph is a surface of revolution about the $x$ 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 $y$ axis. It shows that the graph's surface for positive and negative $y$ values doesn't really meet along the negative real $v$ axis, but instead forms a spiral surface about the $y$ axis. Because its $y$ values have been extended to , this image also better depicts the 2π periodicity in the imaginary $y$ value.

Computation of where both and are complex

Complex exponentiation can be defined by converting to polar coordinates and using the identity :
$a^b = \left(re^\right)^b = \left(e^\right)^b = e^$

However, when is not an integer, this function is multivalued, because is not unique (see failure of power and logarithm identities).

Matrices and Banach algebras

The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra . In this setting, , and is invertible with inverse for any in . If , then , but this identity can fail for noncommuting and .

Some alternative definitions lead to the same function. For instance, can be defined as
$\lim_ \left(1 + \frac \right)^n .$

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

Lie algebras

Given a Lie group and its associated Lie algebra $\mathfrak$, the exponential map is a map $\mathfrak$ satisfying similar properties. In fact, since is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group of invertible matrices has as Lie algebra , the space of all matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

The identity can fail for Lie algebra elements and that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.

Transcendency

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

If are distinct complex numbers, then are linearly independent over . It follows that is transcendental over .

Computation

When computing (an approximation of) the exponential function near the argument , the result will be close to 1, and computing the value of the difference $e^x-1$ with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large calculation error, possibly even a meaningless result.

Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1, for computing directly, bypassing computation of . For example, if the exponential is computed by using its Taylor series
$e^x = 1 + x + \frac 2 + \frac6 + \cdots + \frac + \cdots,$
one may use the Taylor series of $e^x-1$:
$e^x-1=x+\frac 2 + \frac6+\cdots +\frac+\cdots.$

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: $2^x - 1$ and $10^x - 1$.

A similar approach has been used for the logarithm (see lnp1).

An identity in terms of the hyperbolic tangent,
$\operatorname (x) = e^x - 1 = \frac,$
gives a high-precision value for small values of on systems that do not implement .

Alternatively, this expression can be used:

: $e^x - 1 = \lim_ \frac \sum_^n \left(1 + \frac \right)^$

See also

* Carlitz exponential, a characteristic analogue
*
*
* Gaussian function
* Half-exponential function, a compositional square root of an exponential function
* List of exponential topics
* List of integrals of exponential functions
* Mittag-Leffler function, a generalization of the exponential function
* -adic exponential function
* Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions
*

Notes

References

External links

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Elementary special functions
Analytic functions
Exponentials
Special hypergeometric functions
E (mathematical constant)