Exponential Function

at that point.]]

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''.

Sometimes, especially in the Science s, 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 .

PROPERTIES

Most simply, exponential functions multiply at a constant rate. For example the population of a bacterial culture which doubles every 20 minutes can (approximatively, as this is not really a continuous problem) be expressed as an exponential, as can the value of a car which decreases by 10% per year.

Using the natural logarithm, one can define more general exponential functions. The function
:

\,\!\, a^x=(e^{\ln a})^x=e^{x \ln a}

defined for all ''a'' > 0, and all real numbers ''x'', is called the exponential function with base '''''a'''''. Note that this definition of

\, a^x

rests on the previously established existence of the function

\, e^x

, defined for all real numbers. (Here, we neither formally nor conceptually clarify whether such a function exists or what non-natural exponents are supposed to mean.)

Note that the equation above holds for ''a'' = ''e'', since
:

\,\!\, e^{x \ln e}=e^{x \cdot 1}=e^x.

Exponential functions "translate between addition and multiplication" as is expressed in the first three and the fifth of the following ''exponential laws'':
:

\,\!\, a^0 = 1

\,\!\, a^1 = a

\,\!\, a^{x + y} =  a^x a^y

\,\!\, a^{x y} = \left( a^x 
ight)^y

\,\!\, {1 \over a^x} = \left({1 \over a}
ight)^x = a^{-x}

\,\!\, a^x b^x = (a b)^x

These are valid for all positive real numbers ''a'' and ''b'' and all real numbers ''x'' and ''y''. Expressions involving Fraction s and Roots can often be simplified using exponential notation:
:

\,{1 \over a} = a^{-1}

and, for any ''a'' > 0, real number ''b'', and integer ''n'' > 1:
:

{a} ight)^b = a^{b/n}.

DERIVATIVES AND DIFFERENTIAL EQUATIONS

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

:

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

That is, ''e''^''x'' is its own Derivative . 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 the 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)}

.

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

or as the Limit Of A Sequence :

:

e^x = \lim_{n 	o \infty} \left( 1 + {x \over n} 
ight)^n.

In these definitions, ''n''! stands for the Factorial of ''n'', and ''x'' can be any Real Number , Complex Number , element of a Banach Algebra (for example, a Square Matrix ), or member of the field of ''p''-adic Numbers .

For further explanation of these definitions and a proof of their equivalence, see the article Characterizations Of The Exponential Function .

NUMERICAL VALUE

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

:

\,e^x = {1 \over 0!} + x \, \left( {1 \over 1!} + x \, \left( {1 \over 2!} + x \, \left( {1 \over 3!} + \cdots 
ight)
ight)
ight)

\,= 1 + {x \over 1} \left(1 + {x \over 2} \left(1 + {x \over 3} \left(1 + \cdots 
ight)
ight)
ight)

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

To ensure this, we can use the following identity.

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'' 2^''n'' 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) + n\ln(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 = \left\lfloorrac{x}{\ln(2)}
ight
floor.

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

:

\,u = x - n\ln(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 + u(1 + u(rac{1}{2!} + u(rac{1}{3!} + u(....)))).

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''<SUP>''X''</SUP>

Via Euler's identity:

:

\,
\ e^x=1+x+rac{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}}}}}}\,

COMPUTATION OF <MATH>\,A^N</MATH> FOR NATURAL NUMBER (POSITIVE INTEGER) ''N''

There is a fast way to compute

\,a^n

when ''n'' is a positive integer. It makes use of the fact that testing that such a number is odd is very easy on a computer and dividing by 2 is also fast by simply shifting all the bits to the right.

step 1, initialize some variables

y := 1, k := n, f := a

step 2, test k

if k is 0, '''go to''' step 7

step 3, (k is not 0 here, test if k is even)

if k is even '''go to''' step 5

step 4, (k is odd here, multiply in)

step 5, (divide k by 2 / ignore remainder, divide by shift, also square f)

k := k shift right by 1

step 6, (loop)

go back to step 2

step 7, (done, y is result = aⁿ)

return y

In C you can write the algorithm like this:



   double power(double a, unsigned int n)

   {

        double y = 1;

        double f = a;

        unsigned int k = n;

        while (k != 0) {


= f;


           k >>= 1;




= f;


        }



        return y;

   }

While a naive multiplication of a^100 would require 100 iterations of a loop multiplying a, this loop iterates only 7 times (The number 100 is written using 7 bits).

This algorithm can easily be extended for signed integers by doing the following steps before and after:

step 1. if k is negative, negate the value so we get a positive k. n still remembers the original value.

	CATEGORIES ABOUT EXPONENTIAL FUNCTION

	elementary special functions

	complex analysis

	exponentials

	special hypergeometric functions

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Information About

Exponential Function