Exponential function

The exponential function $y=e^{x}$

In mathematics, the exponential function is the function e^x, where e is the number (approximately 2.718281828) such that the function e^x equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable. The function is often written as exp(x), especially when the input is an expression too complex to be written as an exponent.

Exponential Function
Representation	$e^{x}\,$
Inverse	$\ln x\,$
Derivative	$e^{x}\,$
Indefinite Integral	$e^{x}+C\,$

The graph of y = e^x is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote. The slope of the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some older sources refer to the exponential function as the anti-logarithm.

Sometimes the term exponential function is used more generally for functions of the form cb^x, where the base b is any positive real number, not necessarily e. See exponential growth for this usage.

In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below.

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 that led Jacob Bernoulli in 1683 to the number

$\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n},$

now known as e. 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 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,

$\,\exp(x)=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n},$

first given by 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 which is why it can be written as e^x

$\exp(x+y)=\exp(x)\cdot \exp(y).$

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. 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. This function property leads to exponential growth and 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.

Formal definition

Main article: Characterizations of the exponential function

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

The exponential function e^x can be characterized in a variety of equivalent ways. In particular it may be defined by the following power series:

$e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+\cdots .$

Using an alternate definition for the exponential function leads to the same result when expanded as a Taylor series.

Less commonly, e^x is defined as the solution y to the equation

$x=\int _{1}^{y}{dt \over t}.$

It is also the following limit:

$e^{x}=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.$

The error term of this limit-expression is described by

$\left(1+{\frac {x}{n}}\right)^{n}=e^{x}\left(1-{\frac {x^{2}}{2n}}+{\frac {x^{3}(8+3x)}{24n^{2}}}+\cdots \right),$

where, the polynomial's degree (in x) in the term with denominator n^k is 2k.

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), if a tangent line (red), and a vertical line (green) are drawn as shown, then the triangle these lines make with the x-axis will have a base (green) of length 1. So the slope of the tangent line (the derivative)at that point P is equal to the height of the triangle (the value of the function).

The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,

$\,{d \over dx}e^{x}=e^{x}.$

That is, e^x is its own derivative and hence is a simple example of a pfaffian function. Functions of the form ke^x for constant k are the only functions with that property. (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.

In fact, many differential equations give rise to exponential functions, including the Schrödinger equation and Laplace's equation as well as the equations for simple harmonic motion.

For exponential functions with other bases:

${d \over dx}a^{x}=(\ln a)a^{x}.$

Thus, any exponential function is a constant multiple of its own derivative.

For exponential functions with other bases and a constant in the exponent:

$\left(c^{ax}\right)'={c^{ax}\ln c\cdot a},\qquad c>0$

note that the equation above is true for all c, but the derivative for c < 0 yields a complex number.

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.

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

${d \over dx}e^{f(x)}=f'(x)e^{f(x)}.$

Continued fractions for e^x

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

$\,\ e^{x}=1+{\cfrac {x}{1-{\cfrac {x}{x+2-{\cfrac {2x}{x+3-{\cfrac {3x}{x+4-{\cfrac {4x}{x+5-{\cfrac {5x}{x+6-\ddots }}}}}}}}}}}}$

The following generalized continued fraction for e^2x/y converges more quickly:

$e^{2x/y}=1+{\cfrac {2x}{y-x+{\cfrac {x^{2}}{3y+{\cfrac {x^{2}}{5y+{\cfrac {x^{2}}{7y+{\cfrac {x^{2}}{9y+{\cfrac {x^{2}}{11y+{\cfrac {x^{2}}{13y+\ddots \,}}}}}}}}}}}}}}$

with a special case for x = y = 1:

$e^{2}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+{\cfrac {1}{13+\ddots .}}}}}}}}}}$

Complex plane

Exponential function on the complex plane. The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right. The periodic horizontal bands indicate that the exponential function is periodic in the imaginary part of its argument.

As in the real case, the exponential function can be defined on the complex plane in several equivalent forms. Some of these definitions mirror the formulas for the real-valued exponential function. Specifically, one can still use the power series definition, where the real value is replaced by a complex one:

$\,\!\,e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}$

Using this definition, it is easy to show why ${d \over dz}e^{z}=e^{z}$ holds in the complex plane.

Another definition extends the real exponential function. First, we state the desired property $e^{x+iy}=e^{x}e^{iy}$. For e^x we use the real exponential function. We then proceed by defining only: $e^{iy}=\cos(y)+i\sin(y)$. Thus we use the real definition rather than ignore it.

When considered as a function defined on the complex plane, the exponential function retains the important properties

$\,\!\,e^{z+w}=e^{z}e^{w}$

$\,\!\,e^{0}=1$

$\,\!\,e^{z}\neq 0$

$\,\!\,{d \over dz}e^{z}=e^{z}$

for all z and w.

The exponential function is an entire function as it is holomorphic over the whole complex plane. It takes on every complex number as value, except the value 0. This is an example of Picard's little theorem that any non-constant entire function takes on every complex number as value, except possibly one.

The exponential function is periodic with imaginary period $2\pi i$ and can be written as

$\,\!\,e^{a+bi}=e^{a}(\cos b+i\sin b)$

where a and b are real values. This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.

See also Euler's formula.

Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). We can then define a more general exponentiation:

$\,\!\,z^{w}=e^{w\ln z}$

for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions. Because it is multi-valued the rule about multiplying exponents for positive real numbers doesn't work in general:

$\,\!\,(e^{z})^{w}\neq e^{\left(zw\right)}$

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 might be noted: 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.

• Plots of the exponential function on the complex plane
• 
  z = Re(e^x+iy)
• 
  z = Im(e^x+iy)
• 

Computation of e^z for a complex z

If $z=x+yi$, where x and y are real numbers, then

$\,e^{z}=e^{x}e^{yi}=e^{x}(\cos y+i\sin y)=e^{x}\cos y+ie^{x}\sin y.$

Computation of a^b where both a and b are complex

Main article: Exponentiation

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

$\,a^{b}=(re^{{\theta }i})^{b}=(e^{\ln(r)+{\theta }i})^{b}=e^{(\ln(r)+{\theta }i)b}.$

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 Banach algebra B. In this setting, e⁰ = 1, and e^x is invertible with inverse e^−x for any x in B. If xy =yx, then e^x+y = e^xe^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 $\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$. Or e^x can be defined as f(1), where f: R→B is the solution to the differential equation f′(t) = xf(t) with initial condition f(0) = 1.

On Lie algebras

Given a Lie group G and its associated Lie algebra ${\mathfrak {g}}$, the exponential map is a map ${\mathfrak {g}}\to G$ 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.

Double exponential function

Main article: double exponential function

The term double exponential function can have two meanings:

• a function with two exponential terms, with different exponents
• a function f(x) = a^ax; this grows even faster than an exponential function; for example, if a = 10: f(−1) = 1.26, f(0) = 10, f(1) = 10¹⁰, f(2) = 10¹⁰⁰ = googol, ..., f(100) = googolplex.

Factorials grow faster than exponential functions, but slower than double-exponential functions. Fermat numbers, generated by $\,F(m)=2^{2^{m}}+1$ and double Mersenne numbers generated by $\,MM(p)=2^{(2^{p}-1)}-1$ are examples of double exponential functions.

Similar properties of e and the function e^z

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 {a₁,..., a_n}, the set {e^a₁z,..., e^a_nz} is linearly independent over C(z).

The function e^z is transcendental over C(z).

See also

• e (mathematical constant)
• Characterizations of the exponential function
• Exponential field
• Exponential growth
• Exponentiation
• List of exponential topics
• List of integrals of exponential functions
• p-adic exponential function
• Tetration

External links

• "Complex exponential function". PlanetMath.
• "Derivative of exponential function". PlanetMath.
• Derivative of exponential function interactive graph
• Weisstein, Eric W. "Exponential Function". MathWorld.
• Complex Exponential Function Module by John H. Mathews
• Taylor Series Expansions of Exponential Functions at efunda.com
• Complex exponential interactive graphic