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 ex, 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.
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. Its y value always equals the slope at that point.As a function of the real variable x, the graph of y = ex 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 cbx, 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.
Contents [hide]
1 Overview and motivation
2 Formal definition
3 Derivatives and differential equations
4 Continued fractions for ex
5 On the complex plane
6 Computation of ez for a complex z
7 Computation of ab where both a and b are complex
8 Matrices and Banach algebras
9 On Lie algebras
10 Double exponential function
11 Similar properties of e and the function ez
12 See also
13 References
14 External links
[edit] Overview and motivation
The exponential function is written as an exponentiation of the mathematical constant e because it is equal to e when applied to 1 and obeys the basic exponentiation identity, that is:
It is the unique continuous function satisfying these identities for real number exponents. Because of this it can be used to define exponentiation to a non rational exponent.
The exponential function has an analytic continuation which is an entire function, that is it has no singularity over the whole complex plane. The occurrence of the exponential function in Euler's formula gives it a central place when working with complex numbers. The definition has been usefully extended to some non-numeric exponents, for instance as the matrix exponential or the exponential map.
There are a number of other characterizations of the exponential function. The one which mainly leads to its pervasive use in mathematics is as the function for which the rate of change is equal to its value, and which is 1 at 0. In the general case where the rate of change is directly proportional (rather than equal) to the value the resulting function can be expressed using the exponential function as follows:
gives
If b = ek then this has the form cbx. Exponentiation with a general base b as in bx (called the exponential function with base b) is defined using the exponential function and its inverse the natural logarithm as follows:
Its use in science is described in exponential growth and exponential decay.
[edit] 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 ex can be defined, in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series:
.
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, ex is defined as the solution y to the equation
It is also the following limit:
The error term of this limit-expression is described by
where, the polynomial's degree (in x) in the nominator with denominator nk is 2k.
[edit] Derivatives and differential equations
The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,
That is, ex is its own derivative and hence is a simple example of a pfaffian function. Functions of the form Kex for constant K are the only functions with that property. (This follows from the Picard-Lindelöf theorem, with y(t) = et, 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 d