Exponential function: Definition from Answers.com

Part of a series of articles on
The mathematical constant e

Natural logarithm · Exponential function

Applications in: compound interest · Euler's identity & Euler's formula · half-lives & exponential growth/decay

Defining e: proof that e is irrational · representations of e · Lindemann–Weierstrass theorem

The exponential function is a function in mathematics that produces as its output the exponentiation of Euler's number (e) by a real input variable. 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 the mathematical constant that is the base of the natural logarithm (approximately 2.71828182846) and that 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. In older sources it is often referred as an anti-logarithm which is the inverse function of a logarithm.

Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form cb^x, where b, called the base, 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.

1 Formal definition
2 Derivatives and differential equations
3 Continued fractions for e^x
4 On the complex plane
5 Computation of e^z for a complex z
6 Computation of a^b where both a and b are complex
7 Matrices and Banach algebras
8 On Lie algebras
9 Double exponential function
10 Similar properties of e and the function e^z
11 See also
12 References
13 External links

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 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:

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

The error term of this limit-expression is described by

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

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

If b = e^k then this has the form cb^x. Exponentiation with a general base b as in b^x (called the exponential function with base b) is defined using the exponential function and its inverse the natural logarithm as follows:

$\,\!\, b^x=\left(e^{\ln b}\right)^x=e^{x \ln b}.\,$

Exponential functions arise in problems involving proportional growth or decay of quantities. The use of such functions in science is described in exponential growth and exponential decay.

Derivatives and differential equations

The derivative of the exponential function is equal to the value of the function. From any point on the function (blue), if a tangent line (red), and a vertical line (grey) are drawn as shown, then the triangle these lines make with the x-axis will have a base (green) of length 1. Then the slope of the tangent line line(the derivative) is equal to the height of the triangle(the value of the function).

The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0. The slope of the graph at each point is equal to its y co-ordinate at that point.]] 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.

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:

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

The following generalized continued fraction for $e x$ converges more quickly:

$\, \ e^x=1+\cfrac{2x}{(2-x)+\cfrac{x^2}{6+\cfrac{x^2}{10+\cfrac{x^2}{14+\cfrac{x^2}{18+\cfrac{x^2}{22+\ddots}}}}}}\,$

On the 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 $e x + i y = e x e i y$ . For $e x$ we use the real exponential function. We then proceed by defining only: $e i y = c o s (y) + i s i n (y)$ . Thus we use the real definition rather than ignore it.^[1]

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.

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

Computation of e^z for a complex z

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

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

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 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

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

$\,\ f(0) = 1$

$\,\ f'(t) = A f(t)$

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

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^{a^x}; 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).

References

^ Ahlfors, Lars V. (1953). Complex analysis. McGraw-Hill Book Company, Inc..

External links

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