Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp(x). Equivalently, this can be written in the form e^x, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828, and is also known as Euler's number.

As a function of the real variable x, the graph of y=e^x is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x. The exponential function is occasionally referred to as the anti-logarithm. However, this terminology seems to have fallen into disuse in recent times.

Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form ka^x, where a, called the base, is any positive real number not equal to one. This article will focus initially on the exponential function with base e, Euler's number.

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 and motivation

The importance of the exponential function in the physical sciences stems from its use in mathematical models of exponential growth and exponential decay. If a quantity y depends on another quantity x in such a way that y is multiplied by a constant positive factor a whenever x increases by 1 unit then:

$y=ca^x,\; ,$

where c is the value of y when x is 0. The general exponential function a^x (called the exponential function with base a) is defined using the natural logarithm as follows:

$,!,\; a^x=left(e^\{ln\; a\}right)^x=e^\{x\; ln\; a\},\; ,$

defined for all a > 0, and all real numbers x. Note that once the existence of the function e^x has been established for all real numbers, then a^x is defined for all positive values of a.

The exponential function is used to model many physical scenarios, including radioactive decay; unchecked population growth (the Malthusian growth model); DC current in an RC circuit; and heat transfer between two bodies (Newton's law of cooling).

Derivatives and differential equations

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. (This follows from the Picard-Lindelöf theorem, with y(t) = e^t, y(0)=K and f(t,y(t)) = y(t).) 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:

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

A proof being,

$y\; =\; a^x$

$ln\; y\; =\; ln\{a^x\}$

$ln\; y\; =\; x\; ln\; a$

$frac\{1\}\{y\}\; frac\{dy\}\{dx\}\; =\; ln\; a$

$frac\{dy\}\{dx\}\; =\; (ln\; a)\; y\; =\; (ln\; a)\; a^x$

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

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:

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

Formal definition

The exponential function e^x can be defined, in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series:

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

Note that this definition has the form of a Taylor series. Using an alternate definition for the exponential function should lead 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:

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

Numerical value

To obtain the numerical value of the exponential function, the infinite series can be rewritten as :

&= 1 + {x over 1} left(1 + {x over 2} left(1 + {x over 3} left(1 + cdots right)right)right).end{align}

This expression will converge quickly if we can ensure that x is less than one.

To ensure this, we can use the following identity.

&= e^z times left[{1 over 0!} + f , left({1 over 1!} + f , left({1 over 2!} + f , left({1 over 3!} + cdots right)right)right)right]end{align}

• Where z is the integer part of x ;
• where f is the fractional part of x ;
• hence, f is always less than 1 and f and z add up to x.

The value of the constant e^z can be calculated beforehand by multiplying e with itself z times.

Computing exp(x) for real x

An even better algorithm can be found as follows.

First, notice that the answer y = e^x is usually a floating point number represented by a mantissa m and an exponent n so y = m 2ⁿ for some integer n and suitably small m. Thus, we get:

$,y\; =\; m,2^n\; =\; e^x.$

Taking log on both sides of the last two gives us:

$,ln(y)\; =\; ln(m)\; +\; nln(2)\; =\; x.$

Thus, we get n as the result of dividing x by log(2) and finding the greatest integer that is not greater than this - that is, the floor function:

$,n\; =\; leftlfloorfrac\{x\}\{ln(2)\}rightrfloor.$

Having found n we can then find the fractional part u like this:

$,u\; =\; x\; -\; nln(2).$

The number u is small and in the range 0 ≤ u < ln(2) and so we can use the previously mentioned series to compute m:

$,m\; =\; e^u\; =\; 1\; +\; uleft(1\; +\; uleft(frac\{1\}\{2!\}\; +\; uleft(frac\{1\}\{3!\}\; +\; uleft(....right)right)right)right).$

Having found m and n we can then produce y by simply combining those two into a floating point number:

$,y\; =\; e^x\; =\; m,2^n.$

Continued fractions for e^x

Via Euler's identity:

$,$

e^x=1+x+frac{x^2}{2!}+cdots= 1+cfrac{x}{1-cfrac{x}{x+2-cfrac{2x}{x+3-cfrac{3x}{x+4-cfrac{4x}{x+5-cfrac{5x}{ddots}}}}}}

More advanced techniques are necessary to construct the following:

$,$

e^{2m/n}=1+cfrac{2m}{(n-m)+cfrac{m^2}{3n+cfrac{m^2}{5n+cfrac{m^2}{7n+cfrac{m^2}{9n+cfrac{m^2}{ddots}}}}}},

Setting m = x and n = 2 yields

$,$

e^x=1+cfrac{2x}{(2-x)+cfrac{x^2}{6+cfrac{x^2}{10+cfrac{x^2}{14+cfrac{x^2}{18+cfrac{x^2}{ddots}}}}}},

On the complex plane

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:

$,!,\; e^z\; =\; sum\_\{n\; =\; 0\}^inftyfrac\{z^n\}\{n!\}$

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 $e^\{x\; +\; iy\}\; =\; e^x\; e^\{i\; y\}$. For $e^x$ we use the real exponential function. We then proceed by defining only: $e^\{i\; y\}\; =\; cos(y)\; +\; i\; sin(y)$. 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

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

$,!,\; e^0\; =\; 1$

$,!,\; e^z\; ne\; 0$

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

for all z and w.

It is a holomorphic function which 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:

$,!,\; (e^z)^w\; ne\; e^left(z\; wright)$

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.

Computation of exp(z) for a complex z

This is fairly straightforward given the formula

$,e^\{x\; +\; yi\}\; =\; e^xe^\{yi\}\; =\; e^x(cos(y)\; +\; i\; sin(y))\; =\; e^xcos(y)\; +\; ie^xsin(y).$

Note that the argument y to the trigonometric functions is real.

Computation of 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:

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

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 definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential). In this case we have

$,\; e^\{x\; +\; y\}\; =\; e^x\; e^y\; mbox\{\; if\; \}\; xy\; =\; yx$

$,\; e^0\; =\; 1$

$,\; e^x$ is invertible with inverse $,\; e^\{-x\}$

the derivative of $,\; e^x$ at the point $,\; x$ is that linear map which sends $,\; u$ to $,\; ue^x$.

In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:

$,\; f(t)\; =\; e^\{t\; A\}$

where A is a fixed element of the algebra and t is any real number. This function has the important properties

$,\; f(s\; +\; t)\; =\; f(s)\; f(t)$

$,\; f(0)\; =\; 1$

$,\; f\text{'}(t)\; =\; A\; f(t)$

On Lie algebras

The exponential map sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M (n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map. In general, when the argument of the exponential function is noncommutative, the formula is given explicitly by the Baker-Campbell-Hausdorff formula.

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) (ie. not the quotient of two polynomials with complex coefficients).

For n distinct complex numbers {a₁,..., a_n}, {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
• Tetration
• Exponential growth
• Exponentiation
• List of integrals of exponential functions
• List of exponential topics

References

External links

• Complex Exponential Function Module by John H. Mathews
• Taylor Series Expansions of Exponential Functions at efunda.com
• Complex exponential interactive graphic