Exponential function

In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. The exponential of a variable Template:Tmath is denoted Template:Tmath or Template:Tmath, with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant [[e (mathematical constant)|number Template:Math]], 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 Template:Math to the multiplicative identity Template:Math, and the exponential of a sum is equal to the product of separate exponentials, Template:Tmath. Its inverse function, the natural logarithm, Template:Tmath or Template:Tmath, converts products to sums: Template:Tmath.

The exponential function is occasionally called the natural exponential function, matching the name natural logarithm, for distinguishing it from some other functions that are also commonly called exponential functions. These functions include the functions of the form Template:Tmath, which is exponentiation with a fixed base Template:Tmath. More generally, and especially in applications, functions of the general form Template:Tmath are also called exponential functions. They grow or decay exponentially in that the rate that Template:Tmath changes when Template:Tmath is increased is proportional to the current value of Template:Tmath.

The exponential function can be generalized to accept complex numbers as arguments. This reveals relations between multiplication of complex numbers, rotations in the complex plane, and trigonometry. Euler's formula Template:Tmath expresses and summarizes these relations.

The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras.

Graph

The graph of $y=e^x$ is upward-sloping, and increases faster than every power of Template:Tmath. The graph always lies above the Template:Mvar-axis, but becomes arbitrarily close to it for large negative Template:Mvar; thus, the Template:Mvar-axis is a horizontal asymptote. The equation $\frac{d}{dx}{e}^x=e^x$ means that the slope of the tangent to the graph at each point is equal to its height (its Template:Mvar-coordinate) at that point.

Definitions and fundamental properties

There are several equivalent definitions of the exponential function, although of very different nature.

Differential equation

File:Exp tangent.svg

The derivative of the exponential function is equal to the value of the function. Since the derivative is the slope of the tangent, this implies that all green right triangles have a base length of 1.

One of the simplest definitions is: The exponential function is the unique differentiable function that equals its derivative, and takes the value Template:Math for the value Template:Math 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 Template:Tmath and Template:Tmath are two functions satisfying the above definition, then the derivative of Template:Tmath is zero everywhere because of the quotient rule. It follows that Template:Tmath is constant; this constant is Template:Math since Template:Tmath.

Existence is proved in each of the two following sections.

Inverse of natural logarithm

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

$\begin{array}{rl}\ln (\exp x)& =x\\ \exp (\ln y)& =y\end{array}$

for every real number $x$ and every positive real number $y.$

Power series

The exponential function is the sum of the power series $$\begin{array}{rl}\exp (x)& =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \\ & =\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!},\end{array}$$

File:Exp series.gif

The exponential function (in blue), and the sum of the first Template:Math terms of its power series (in red)

where $n!$ is the factorial of Template:Mvar (the product of the Template:Mvar first positive integers). This series is absolutely convergent for every $x$ per the ratio test. So, the derivative of the sum can be computed by term-by-term differentiation, 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 Template:Tmath, and is everywhere the sum of its Maclaurin series.

Functional equation

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

It can be proved that a function that satisfies this functional equation has the form Template:Tmath if it is either continuous or monotonic. It is thus differentiable, and equals the exponential function if its derivative at Template:Math is Template:Math.

Limit of integer powers

The exponential function is the limit, as the integer Template:Mvar goes to infinity, $$\exp (x)=\underset{n\to +\mathrm{\infty }}{\lim }{(1+\frac{x}{n})}^n.$$ By continuity of the logarithm, this can be proved by taking logarithms and proving $$x=\underset{n\to \mathrm{\infty }}{\lim }\ln {(1+\frac{x}{n})}^n=\underset{n\to \mathrm{\infty }}{\lim }n\ln (1+\frac{x}{n}),$$ for example with Taylor's theorem.

Properties

Reciprocal: The functional equation implies Template:Tmath. Therefore Template:Tmath for every Template:Tmath and $$\frac{1}{e^x}=e^{-x}.$$

Positiveness: Template:Tmath for every real number Template:Tmath. This results from the intermediate value theorem, since Template:Tmath and, if one would have Template:Tmath for some Template:Tmath, there would be an Template:Tmath such that Template:Tmath between Template:Tmath and Template:Tmath. Since the exponential function equals its derivative, this implies that the exponential function is monotonically increasing.

Extension of exponentiation to positive real bases: Let Template:Mvar be a positive real number. The exponential function and the natural logarithm being the inverse each of the other, one has $b=\exp (\ln b).$ If Template:Mvar is an integer, the functional equation of the logarithm implies $$b^n=\exp (\ln b^n)=\exp (n\ln b).$$ Since the right-most expression is defined if Template:Mvar is any real number, this allows defining Template:Tmath for every positive real number Template:Mvar and every real number Template:Mvar: $$b^x=\exp (x\ln b).$$ In particular, if Template:Mvar is the Euler's number $e=\exp (1),$ one has $\ln e=1$ (inverse function) and thus $$e^x=\exp (x).$$ This shows the equivalence of the two notations for the exponential function.

General exponential functions

A function is commonly called an exponential functionTemplate:Mdashwith an indefinite articleTemplate:Mdashif it has the form Template:Tmath, that is, if it is obtained from exponentiation by fixing the base and letting the exponent vary.

More generally and especially in applied contexts, the term exponential function is commonly used for functions of the form Template:Tmath. This may be motivated by the fact that, if the values of the function represent quantities, a change of measurement unit changes the value of Template:Tmath, and so, it is nonsensical to impose Template:Tmath.

These most general exponential functions are the differentiable functions that satisfy the following equivalent characterizations.

• Template:Tmath for every Template:Tmath and some constants Template:Tmath and Template:Tmath.
• Template:Tmath for every Template:Tmath and some constants Template:Tmath and Template:Tmath.
• The value of $f^{\prime }(x)/f(x)$ is independent of $x$.
• For every $d,$ the value of $f(x+d)/f(x)$ is independent of $x;$ that is, $$\frac{f(x+d)}{f(x)}=\frac{f(y+d)}{f(y)}$$ for every Template:Mvar, Template:Mvar.

File:Exponenciala priklad.png

Exponential functions with bases 2 and 1/2

The base of an exponential function is the base of the exponentiation that appears in it when written as Template:Tmath, namely Template:Tmath. The base is Template:Tmath in the second characterization, $\exp \frac{f^{\prime }(x)}{f(x)}$ in the third one, and ${\left(\frac{f(x+d)}{f(x)}\right)}^{1/d}$ in the last one.

In applications

The last characterization is important in empirical sciences, as allowing a direct experimental test whether a function is an exponential function.

Exponential growth or exponential decayTemplate:Mdashwhere the variable change is proportional to the variable valueTemplate:Mdashare thus modeled with exponential functions. Examples are unlimited population growth leading to Malthusian catastrophe, continuously compounded interest, and radioactive decay.

If the modeling function has the form Template:Tmath or, equivalently, is a solution of the differential equation Template:Tmath, the constant Template:Tmath is called, depending on the context, the decay constant, disintegration constant, rate constant, or transformation constant.

Equivalence proof

For proving the equivalence of the above properties, one can proceed as follows.

The two first characterizations are equivalent, since, if Template:Tmath and Template:Tmath, one has $$e^{kx}=(e^k{)}^x=b^x.$$ The basic properties of the exponential function (derivative and functional equation) implies immediately the third and the last condition.

Suppose that the third condition is verified, and let Template:Tmath be the constant value of $f^{\prime }(x)/f(x).$ Since $\frac{\partial e^{kx}}{\partial x}=k{e}^{kx},$ the quotient rule for derivation implies that $$\frac{\partial }{\partial x}\frac{f(x)}{e^{kx}}=0,$$ and thus that there is a constant Template:Tmath such that $f(x)=a{e}^{kx}.$

If the last condition is verified, let $\phi (d)=f(x+d)/f(x),$ which is independent of Template:Tmath. Using Template:Tmath, one gets $$\frac{f(x+d)-f(x)}{d}=f(x)\frac{\phi (d)-\phi (0)}{d}.$$ Taking the limit when Template:Tmath tends to zero, one gets that the third condition is verified with Template:Tmath. It follows therefore that Template:Tmath for some Template:Tmath and Template:Tmath As a byproduct, one gets that $${\left(\frac{f(x+d)}{f(x)}\right)}^{1/d}=e^k$$ is independent of both Template:Tmath and Template:Tmath.

Compound interest

The earliest occurrence of the exponential function was in Jacob Bernoulli's study of compound interests in 1683. This is this study that led Bernoulli to consider the number $$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n$$ now known as Euler's number and denoted Template:Tmath.

The exponential function is involved as follows in the computation of continuously compounded interests.

If a principal amount of 1 earns interest at an annual rate of Template:Math compounded monthly, then the interest earned each month is Template:Math times the current value, so each month the total value is multiplied by Template:Math, and the value at the end of the year is Template:Math. If instead interest is compounded daily, this becomes Template:Math. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, $$\exp x=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{x}{n})}^n$$ first given by Leonhard Euler.

Differential equations

Exponential functions occur very often in solutions of differential equations.

The exponential functions can be defined as solutions of differential equations. Indeed, the exponential function is a solution of the simplest possible differential equation, namely Template:Tmath. Every other exponential function, of the form Template:Tmath, is a solution of the differential equation Template:Tmath, and every solution of this differential equation has this form.

The solutions of an equation of the form $$y^{\prime }+ky=f(x)$$ involve exponential functions in a more sophisticated way, since they have the form $$y=c{e}^{-kx}+e^{-kx}\int f(x)e^{kx}dx,$$ where Template:Tmath is an arbitrary constant and the integral denotes any antiderivative of its argument.

More generally, the solutions of every linear differential equation with constant coefficients can be expressed in terms of exponential functions and, when they are not homogeneous, antiderivatives. This holds true also for systems of linear differential equations with constant coefficients.

Complex exponential

The exponential function Template:Math plotted in the complex plane from Template:Math to Template:Math

The exponential function Template:Math plotted in the complex plane from Template:Math to Template:Math

File:Exp-complex-cplot.svg

A complex plot of $z\mapsto \exp z$, with the argument $\mathrm{Arg}\exp z$ represented by varying hue. The transition from dark to light colors shows that $|\exp z|$ is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that $z\mapsto \exp z$ is periodic in the imaginary part of $z$.

The exponential function can be naturally extended to a complex function, which is a function with the complex numbers as domain and codomain, such that its restriction to the reals is the above-defined exponential function, called real exponential function in what follows. This function is also called the exponential function, and also denoted Template:Tmath or Template:Tmath. For distinguishing the complex case from the real one, the extended function is also called complex exponential function or simply complex exponential.

Most of the definitions of the exponential function can be used verbatim for definiting the complex exponential function, and the proof of their equivalence is the same as in the real case.

The complex exponential function can be defined in several equivalent ways that are the same as in the real case.

The complex exponential is the unique complex function that equals its complex derivative and takes the value Template:Tmath for the argument Template:Tmath: $$\frac{d{e}^z}{dz}=e^z\text{and}{e}^0=1.$$

The complex exponential function is the sum of the series $$e^z=\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{k!}.$$ This series is absolutely convergent for every complex number Template:Tmath. So, the complex differential is an entire function.

The complex exponential function is the limit $$e^z=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{z}{n})}^n$$

As with the real exponential function (see Template:Section link above), the complex exponential satisfies the functional equation $$\exp (z+w)=\exp (z)\cdot \exp (w).$$ Among complex functions, it is the unique solution which is holomorphic at the point Template:Tmath and takes the derivative Template:Tmath there.

The complex logarithm is a right-inverse function of the complex exponential: $$e^{\log z}=z.$$ However, since the complex logarithm is a multivalued function, one has $$\log e^z=\{z+2ik\pi \mid k\in \mathbb{\mathbb{Z}}\},$$ and it is difficult to define the complex exponential from the complex logarithm. On the opposite, this is the complex logarithm that is often defined from the complex exponential.

The complex exponential has the following properties: $$\frac{1}{e^z}=e^{-z}$$ and $$e^z\ne 0\text{for every }z\in \mathbb{ℂ}.$$ It is periodic function of period Template:Tmath; that is $$e^{z+2ik\pi }=e^z\text{for every }k\in \mathbb{\mathbb{Z}}.$$ This results from Euler's identity Template:Tmath and the functional identity.

The complex conjugate of the complex exponential is $$\overline{e^z}=e^{\bar{z}}.$$ Its modulus is $$|e^z|=e^{\mathrm{\Re }(z)},$$ where Template:Tmath denotes the real part of Template:Tmath.

Relationship with trigonometry

Complex exponential and trigonometric functions are strongly related by Euler's formula: $$e^{it}=\cos (t)+i\sin (t).$$

This formula provides the decomposition of complex exponentials into real and imaginary parts: $$e^{x+iy}=e^x{e}^{iy}=e^x\cos y+i{e}^x\sin y.$$

The trigonometric functions can be expressed in terms of complex exponentials: $$\begin{array}{rl}\cos x& =\frac{e^{ix}+e^{-ix}}{2}\\ \sin x& =\frac{e^{ix}-e^{-ix}}{2i}\\ \tan x& =i\frac{1-e^{2ix}}{1+e^{2ix}}\end{array}$$

In these formulas, Template:Tmath are commonly interpreted as real variables, but the formulas remain valid if the variables are interpreted as complex variables. These formulas may be used to define trigonometric functions of a complex variable.

Plots

• 3D plots of real part, imaginary part, and modulus of the exponential function
• Template:Math
  Template:Math
• Template:Math
  Template:Math
• Template:Math
  Template:Math

Considering the complex exponential function as a function involving four real variables: $$v+iw=\exp (x+iy)$$ 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.

• Graphs of the complex exponential function
• Checker board key: x>0:green x<0:red y>0:yellow y<0:blue
  Checker board key:
  $x>0:\text{green}$
  $x<0:\text{red}$
  $y>0:\text{yellow}$
  $y<0:\text{blue}$
• Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
  Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
• Projection into the x {\displaystyle x} , v {\displaystyle v} , and w {\displaystyle w} dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image).
  Projection into the $x$, $v$, and $w$ dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)
• Projection into the y {\displaystyle y} , v {\displaystyle v} , and w {\displaystyle w} dimensions, producing a spiral shape. ( y {\displaystyle y} range extended to ±2π, again as 2-D perspective image).
  Projection into the $y$, $v$, and $w$ dimensions, producing a spiral shape ($y$ range extended to ±2Template:Pi, 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 Template:Math, this image also better depicts the 2π periodicity in the imaginary $y$ 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 Template:Math. In this setting, Template:Math, and Template:Math is invertible with inverse Template:Math for any Template:Math in Template:Math. If Template:Math, then Template:Math, but this identity can fail for noncommuting Template:Math and Template:Math.

Some alternative definitions lead to the same function. For instance, Template:Math can be defined as $$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{x}{n})}^n.$$

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

Lie algebras

Given a Lie group Template:Math and its associated Lie algebra $\mathfrak{𝔤}$, the exponential map is a map $\mathfrak{𝔤}$ Template:Math satisfying similar properties. In fact, since Template:Math 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 Template:Math of invertible Template:Math matrices has as Lie algebra Template:Math, the space of all Template:Math 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 Template:Math and Template:Math that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.

Transcendency

The function Template:Math is a transcendental function, which means that it is not a root of a polynomial over the ring of the rational fractions $\mathbb{ℂ}(z).$

If Template:Math are distinct complex numbers, then Template:Math are linearly independent over $\mathbb{ℂ}(z)$, and hence Template:Math is transcendental over $\mathbb{ℂ}(z)$.

Computation

The Taylor series definition above is generally efficient for computing (an approximation of) $e^x$. However, when computing near the argument $x=0$, 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 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 Template:Math directly, bypassing computation of Template:Math. For example, one may use the Taylor series: $$e^x-1=x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots +\frac{x^n}{n!}+\cdots .$$

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 Template:Math, 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 log1p.

An identity in terms of the hyperbolic tangent, $$\mathrm{expm1}(x)=e^x-1=\frac{2\tanh (x/2)}{1-\tanh (x/2)},$$ gives a high-precision value for small values of Template:Math on systems that do not implement Template:Math.

Continued fractions

The exponential function can also be computed with continued fractions.

A continued fraction for Template:Math can be obtained via an identity of Euler: $$e^x=1+\frac{x}{1-\frac{x}{x+2-\frac{2x}{x+3-\frac{3x}{x+4-\ddots }}}}$$

The following generalized continued fraction for Template:Math, also due to Euler , converges more quickly: $$e^z=1+\frac{2z}{2-z+\frac{z^2}{6+\frac{z^2}{10+\frac{z^2}{14+\ddots }}}}$$

or, by applying the substitution Template:Math: $$e^{\frac{x}{y}}=1+\frac{2x}{2y-x+\frac{x^2}{6y+\frac{x^2}{10y+\frac{x^2}{14y+\ddots }}}}$$ with a special case for Template:Math: $$e^2=1+\frac{4}{0+\frac{2^2}{6+\frac{2^2}{10+\frac{2^2}{14+\ddots }}}}=7+\frac{2}{5+\frac{1}{7+\frac{1}{9+\frac{1}{11+\ddots }}}}$$

This formula also converges, though more slowly, for Template:Math. For example: $$e^3=1+\frac{6}{-1+\frac{3^2}{6+\frac{3^2}{10+\frac{3^2}{14+\ddots }}}}=13+\frac{54}{7+\frac{9}{14+\frac{9}{18+\frac{9}{22+\ddots }}}}$$

See also

• Carlitz exponential, a characteristic Template:Math analogue
• Template:Annotated link
• Template:Annotated link
• Gaussian function
• Half-exponential function, a compositional square root of an exponential function
• Template:Annotated link - Used for solving exponential equations
• List of exponential topics
• List of integrals of exponential functions
• Mittag-Leffler function, a generalization of the exponential function
• [[p-adic exponential function|Template:Math-adic exponential function]]
• Padé table for exponential function – Padé approximation of exponential function by a fraction of polynomial functions
• Phase factor

External links

• Template:Springer