Exponential function
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The exponential function is one of the most important functions in mathematics. It is written as exp(x) or ex, where e equals approximately 2.71828183 and is the base of the natural logarithm.
As a function of the real variable x, the graph of 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.
Sometimes, especially in the sciences, the term exponential function is reserved for functions of the form kax, where a, called the base, is any positive real number. 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.
Properties
Using the natural logarithm, one can define more general exponential functions. The function- [\!\, a^x=e^]
Note that the equation above holds for a = e, since
- [\!\, e^=e^=e^x.]
- [\!\, a^0 = 1]
- [\!\, a^1 = a]
- [\!\, a^ = a^x a^y]
- [\!\, a^ = \left( a^x \right)^y]
- [\!\, = \left(\right)^x = a^]
- [\!\, a^x b^x = (a b)^x]
- [ = a^]
- [\sqrt[n] = \left(\sqrt[n]\right)^b = a^]
Derivatives and differential equations
The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives. In particular,
- [ e^x = e^x]
- 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
For exponential functions with other bases:
- [ a^x = (\ln a) a^x]
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:
- [ e^ = f'(x)e^].
Formal definition
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:
- [e^x = \sum_^ = 1 + x + + + + \cdots]
- [e^x = \lim_ \left( 1 + \right)^n.]
For further explanation of these definitions and a proof of their equivalence, see the article Definitions of the exponential function.
Numerical value
To obtain the numerical value of the exponential function, the infinite series can be rewritten as :
- [e^x = + x \, \left( + x \, \left( + x \, \left( + \cdots \right)\right)\right)]
- [= 1 + \left(1 + \left(1 + \left(1 + \cdots \right)\right)\right)]
To ensure this, we can use the following identity.
[e^x\,] [=e^\,] [= e^z \times \left[ + f , left( + f , left( + f , left( + cdots right)right)right)right]] - 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].
On the complex plane
When considered as a function defined on the complex plane, the exponential function retains the important properties- [\!\, e^ = e^z e^w]
- [\!\, e^0 = 1]
- [\!\, e^z \ne 0]
- [\!\, e^z = e^z]
It is a holomorphic function which is periodic with imaginary period [2 \pi i] and can be written as
- [\!\, e^ = e^a (\cos b + i \sin b)]
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^]
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 sprial never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.
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^ = e^x e^y \mbox xy = yx]
- [\ e^0 = 1]
- [\ e^x] is invertible with inverse [\ e^]
- the derivative of [\ e^x] at the point [\ x] is that linear map which sends [\ u] to [\ ue^x].
- [\ f(t) = e^]
- [\ 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.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^]; 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(3) = 101000, ..., f(100) = googolplex.
See also
- Characterizations of the exponential function
- Exponential growth
- Exponentiation
- List of exponential topics
External links
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