# Encyclopedia

The exponential function y = ex
.In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex equals its own derivative.^ Mathematical functions » Exponential integrals and error functions ¶ .
• Exponential integrals and error functions — mpmath v0.13 documentation 22 January 2010 19:32 UTC mpmath.googlecode.com [Source type: Academic]

^ And the first derivative and first differences of an exponential function are exponential.

^ According to Wikipedia, the exponential function is the function ex, where e is the number such that the function ex equals its own derivative.
• How to get exponential success on your blog 22 January 2010 19:32 UTC www.catswhoblog.com [Source type: General]

[1][2] .The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable.^ Is there an exponential function that never increases or decreases?
• Session 7, Part A: Exploring Exponential Functions 22 January 2010 19:32 UTC www.learner.org [Source type: FILTERED WITH BAYES]

^ The most useful functions for science are exponential functions.
• MathDL | Exponential Functions 2 February 2010 14:34 UTC mathdl.maa.org [Source type: FILTERED WITH BAYES]

^ Model real-world phenomena using exponential functions.
• Investigation 5: Exploring Exponential Functions 22 January 2010 19:32 UTC www.cdli.ca [Source type: Academic]

.The exponential function is often written as exp(x), especially when the input is an expression too complex to be written as an exponent.^ The function exp() returns the exponential of the argument.

^ Pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function.
• exponential@Everything2.com 2 February 2010 14:34 UTC everything2.com [Source type: FILTERED WITH BAYES]

^ It is so important that it is often called the exponential function.
• Logarithmic Functions 22 January 2010 19:32 UTC www.uncwil.edu [Source type: Reference]

.The graph of y = ex is upward-sloping, and increases faster as x increases.^ The slope of the graph is rapidly increasing.
• Modeling exponential growth - Excel - Microsoft Office Online 2 February 2010 14:34 UTC office.microsoft.com [Source type: FILTERED WITH BAYES]

^ If you increase the intrinsic growth rate, the population grows faster and the model shows a much steeper slope.
• Exponential Growth Models 2 February 2010 14:34 UTC act.hdsb.ca [Source type: Academic]

^ The slope increases through time and soon will shoot off the top of a graph.
• Lab 4 - Population Growth 2 February 2010 14:34 UTC www.angelo.edu [Source type: FILTERED WITH BAYES]

.The graph always lies above the x-axis but can get arbitrarily close to it for negative x; thus, the x-axis is a horizontal asymptote.^ The graph of f has a horizontal asymptote at y = 0.
• Graph of Exponential Functions 22 January 2010 19:32 UTC www.analyzemath.com [Source type: Reference]

^ Thus, we say that the x-axis is an asymptote of each graph.

^ This means that the graph will approach the x axis as an asymptote.
• Introduction to Exponential Functions 2 February 2010 14:34 UTC vhcc2.vhcc.edu [Source type: FILTERED WITH BAYES]

.The slope of the graph at each point is equal to its y coordinate at that point.^ The slope of the tangent to the curve at any point is always equal to its y-coordinate.
• exponential functions 22 January 2010 19:32 UTC www.blc.edu [Source type: FILTERED WITH BAYES]

^ To accomplish this, students are asked to search for the base, b, that defines a function f(x)=b x with the property that at any point on the graph, the slope of the tangent line is equal to f(x).
• Texas Instruments Activities Exchange: Math : Algebra II : Exponential Functions 22 January 2010 19:32 UTC education.ti.com [Source type: Academic]

^ From this point we draw a straight line to the right with slope equal to 1.
• 12.3 - Exponential functions 2 February 2010 14:34 UTC www.mathonweb.com [Source type: Academic]

.The inverse function is the natural logarithm ln(x); because of this, some older sources refer to the exponential function as the anti-logarithm.^ Chapter 10: Exponential and Logarithmic Functions .
• Intermediate Algebra - Chapter 10: Exponential and Logarithmic Functions 22 January 2010 19:32 UTC www.mathnotes.com [Source type: General]

^ In general, the inverse of an exponential function with base a is called the logarithm to the base a .
• Basic Maths: Exponentials & Logs - The Inverse of an Exponential Function 22 January 2010 19:32 UTC www.maths.strath.ac.uk [Source type: Reference]

^ Derivative of the natural logarithm function Let y = ln x.
• Lesson 16: Derivatives of Logarithmic and Exponential Functions 22 January 2010 19:32 UTC www.slideshare.net [Source type: Reference]

.Sometimes the term exponential function is used more generally for functions of the form cbx, where the base b is any positive real number, not necessarily e.^ Real roots and exponentials for complex numbers .
• 10. ln(x) and exp(x), Properties of 22 January 2010 19:32 UTC whyslopes.com [Source type: Academic]

^ Let a be a positive real number.

^ We used an unspecified base a to introduce exponential functions.
• Exponential Functions 22 January 2010 19:32 UTC iteso.mx [Source type: Academic]

.See exponential growth for this usage.^ See entire page » What is exponential growth ?
• Exponential Growth - Ask.com 2 February 2010 14:34 UTC www.ask.com [Source type: FILTERED WITH BAYES]

^ Yahoo sees exponential growth in .
• Yahoo sees exponential growth in | Top Industry News, Statistics, Research and Trends | BNET 2 February 2010 14:34 UTC industry.bnet.com [Source type: FILTERED WITH BAYES]

.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.^ Real roots and exponentials for complex numbers .
• 10. ln(x) and exp(x), Properties of 22 January 2010 19:32 UTC whyslopes.com [Source type: Academic]

^ We see natural numbers as representing real, distinct, objects.

^ See Also: four exponentials conjecture , complex exponential function , example of Taylor polynomials for the exponential function , proof of equivalence of formulas for exp , the groups of real numbers , function , example of jump discontinuity , exponential function defined as limit of powers , derivative of exponential function , exponential function never vanishes .
• PlanetMath: exponential function 22 January 2010 19:32 UTC planetmath.org [Source type: FILTERED WITH BAYES]

 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 People John Napier  · Leonhard Euler

## Overview

.The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.^ A comment on exponential rates of decay: .
• An Intuitive Guide To Exponential Functions & e | BetterExplained 2 February 2010 14:34 UTC betterexplained.com [Source type: General]

^ The exp function returns the exponential value.
• Exponential and Logarithmic Functions 22 January 2010 19:32 UTC www.xgc.com [Source type: Reference]

^ This tells us that the rate of change of the function at a point x is proportional to the function.s value at that point.

One such situation is continuously compounded interest, and in fact it was this that led Jacob Bernoulli in 1683[3] to the number
$\lim_{n o \infty} (1+1/n)^n,$
now known as e. .Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.^ Integrating the exponential function , also part of calculus.
• Exponential and Logarithmic Functions 22 January 2010 19:32 UTC www.intmath.com [Source type: FILTERED WITH BAYES]

^ Why study exponential and logarithmic functions?
• Exponential and Logarithmic Functions 22 January 2010 19:32 UTC www.intmath.com [Source type: FILTERED WITH BAYES]

^ Fitting Functions to Data: Linear and Exponential Regression miscellaneous on-line topics for Finite Mathematics and Calculus Applied to the Real World .
• Regression: Fitting Functions to Data 22 January 2010 19:32 UTC people.hofstra.edu [Source type: FILTERED WITH BAYES]

[3]
.If a principal amount of 1 earns interest at an annual rate of x compounded monthly, then the interest earned each month is x/12 times the current value, so each month the total value is multiplied by (1+x/12), and the value at the end of the year is (1+x/12)12.^ If interest is compounded times per year, the equation we use is: .
• Analyzing Exponential and Logarithmic Functions - CK12 - Flexbooks 2 February 2010 14:34 UTC authors.ck12.org [Source type: Reference]

^ Find the amount, A, after 5 years if the interest is compounded quarterly.

^ Compounding more times a year increases the value of m .
• Exponential Functions and Models 22 January 2010 19:32 UTC www.zweigmedia.com [Source type: Academic]

.If instead interest is compounded daily, this becomes (1+x/365)365.^ Interest If an account has an compound interest rate of r per year compounded n times, then an initial deposit of A0 dollars becomes ( r )nt A0 1 + n after t years.
• Lesson 16: Exponential Growth and Decay 2 February 2010 14:34 UTC www.slideshare.net [Source type: Academic]

^ Sometimes an A is used for compound or accumulated interest instead of an S .
• College Algebra Tutorial on Exponential Functions 22 January 2010 19:32 UTC www.wtamu.edu [Source type: General]

^ You put \$1,000 dollars in the bank for five years with daily compounding at 6% annual interest.
• Exponential Functions 2 February 2010 14:34 UTC fym.la.asu.edu [Source type: FILTERED WITH BAYES]

Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function,
$\,\exp(x) := \lim_{n o\infty}\left(1+\frac{x}{n}\right)^{n},$
first given by Euler.[4] .This is one of a number of characterizations of the exponential function; others involve series or differential equations.^ We stated in the section on exponential functions , that exponential functions were one-to-one.
• 4.2 - Logarithmic Functions and Their Graphs 22 January 2010 19:32 UTC www.richland.edu [Source type: FILTERED WITH BAYES]

^ Differentiation of the natural exponential function (e) .
• Differentiation of the natural exponential function (e) - Science Articles 22 January 2010 19:32 UTC www.physicspost.com [Source type: Academic]

^ "The exponential function is one of many functions in mathematics.
• Exponential Functions - Ask.com 22 January 2010 19:32 UTC www.ask.com [Source type: General]

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity which is why it can be written as ex
$\exp(x+y) = \exp(x) \cdot \exp(y).$
.The derivative (rate of change) of the exponential function is the exponential function itself.^ Derivatives of exponential functions - .
• Derivatives of exponential functions Video – 5min.com 22 January 2010 19:32 UTC www.5min.com [Source type: FILTERED WITH BAYES]

^ And the first derivative and first differences of an exponential function are exponential.

^ The only function whose rate of change or derivative is proportional to itself is the exponential function or: .

.More generally, a function with a rate of change proportional to the function itself (rather than equal to it) is expressible in terms of the exponential function.^ This tells us that the rate of change of the function at a point x is proportional to the function.s value at that point.

^ Just as exponential functions increase with rates proportional to the function itself, they can also decrease.

^ If the rate of growth of a quantity is proportional to the quantity itself, then the growth of the quantity will be exponential.
• Bartlett's Law 2 February 2010 14:34 UTC mysite.du.edu [Source type: Original source]

.Explicitly, for any real constant k, a function fRR satisfies f′ = kf if and only if f(x) = cekx for some constant c.^ When we are integrating the Logarithmic and exponential functions we can multiply constants that have no effect on the derivative, they are there only to satisfy initial condition.

^ Any exponentially growing number will eventually grow larger than any other number which grows at only a constant rate for the same amount of time (and will also grow larger than any function which grows only subexponentially ).
• Exponential growth - encyclopedia article - Citizendium 2 February 2010 14:34 UTC en.citizendium.org [Source type: FILTERED WITH BAYES]

^ An exponential function is of the form f ( x ) = a , for some real number a , as long as a > 0.
• Precalculus: Exponential Functions - CliffsNotes 22 January 2010 19:32 UTC www.cliffsnotes.com [Source type: FILTERED WITH BAYES]

.This function property leads to exponential growth and exponential decay.^ Exponential decay: An Exponential Decay Function is a function of the form: .
• Exponential growth and decay made easy 2 February 2010 14:34 UTC classes.tmcc.edu [Source type: Academic]

^ Examples of exponential growth and decay .

^ Examples of exponential growth and decay.

.The exponential function extends to an entire function on the complex plane.^ On a functional equation for the exponential function of a complex variable.
• Complex Exponential Function 22 January 2010 19:32 UTC math.fullerton.edu [Source type: Academic]

^ The exponential function is the entire function defined by .
• Exponential Function -- from Wolfram MathWorld 22 January 2010 19:32 UTC mathworld.wolfram.com [Source type: Academic]

^ Bibliography for the Complex Exponential Function short .
• Complex Exponential Function 22 January 2010 19:32 UTC math.fullerton.edu [Source type: Academic]

.Euler's formula relates its values at purely imaginary arguments to trigonometric functions.^ Note the special base a = 1, as well as the function values at the special arguments x = 1 and x = a .

^ CTRL+SHIFT+A inserts the argument names and parentheses when the insertion point is to the right of a function name in a formula.
• Excel shortcut and function keys - Excel - Microsoft Office Online 22 January 2010 19:32 UTC office.microsoft.com [Source type: General]

^ One attachment shows formulas for how the graphs of exponential functions change based upon the values of a and b.
• ALEX - Alabama Learning Exchange 2 February 2010 14:34 UTC alex.state.al.us [Source type: FILTERED WITH BAYES]

.The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.^ On the zeros of the q-analogue exponential function.
• Complex Exponential Function 22 January 2010 19:32 UTC math.fullerton.edu [Source type: Academic]

^ II. The interval matrix exponential function.
• The Matrix Exponential 22 January 2010 19:32 UTC ecs.fullerton.edu [Source type: Academic]

^ Algebra, Analytic Geometry: The cis or exponential functions Appetizers and Lessons for Mathematics and Reason Thank you for visiting www.whyslopes.com : 1200+ pages .
• Algebra, Analytic Geometry: The cis or exponential functions 22 January 2010 19:32 UTC whyslopes.com [Source type: FILTERED WITH BAYES]

## Formal definition

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.^ There are in fact a variety of ways to define e .
• http://tutorial.math.lamar.edu/classes/calcI/diffexplogfcns.aspx 2 February 2010 14:34 UTC tutorial.math.lamar.edu [Source type: FILTERED WITH BAYES]

^ The exponential distribution function is defined as: .
• Statistics Glossary: E 22 January 2010 19:32 UTC www.statsoft.com [Source type: Academic]

^ We define an exponential function to be any function of the form: y = y 0 · m x .
• 12.3 - Exponential functions 2 February 2010 14:34 UTC www.mathonweb.com [Source type: Academic]

In particular it may be defined by a power series in the form of a Taylor series expansion:
$e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots.$
.Using an alternate definition for the exponential function leads to the same result when expanded as a Taylor series.^ The most useful functions for science are exponential functions.
• MathDL | Exponential Functions 2 February 2010 14:34 UTC mathdl.maa.org [Source type: FILTERED WITH BAYES]

^ Therefore, a modified exponential function is used: .
• Exponential Functions 22 January 2010 19:32 UTC dwb4.unl.edu [Source type: Academic]

^ Graphing exponential functions using transformations.
• Analyzing Exponential and Logarithmic Functions - CK12 - Flexbooks 2 February 2010 14:34 UTC authors.ck12.org [Source type: Reference]

Less commonly, ex 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
$\left(1+\frac{x}{n}\right)^n=e^x \left(1-\frac{x^2}{2n}+\frac{x^3(8+3x)}{24n^2}+\cdots \right),$
where, the polynomial's degree (in x) in the term with denominator nk is 2k.

## Derivatives and differential equations

.
The derivative of the exponential function is equal to the value of the function.
^ The derivative of the exponential function, e t , is equal to the function itself: de t /dt = e t .
• FFT: Complex Numbers 22 January 2010 19:32 UTC www.relisoft.com [Source type: FILTERED WITH BAYES]

^ "The derivative of an exponential function with base a .
• Derivatives of exponential and logarithmic functions - An approach to calculus 2 February 2010 14:34 UTC www.themathpage.com [Source type: FILTERED WITH BAYES]

^ A function that approximates these values is the exponential function, .

From any point on the graph (blue), if a tangent line (red), and a vertical line (green) 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 importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives.^ The most useful functions for science are exponential functions.
• MathDL | Exponential Functions 2 February 2010 14:34 UTC mathdl.maa.org [Source type: FILTERED WITH BAYES]

^ Mathematical functions » Exponential integrals and error functions ¶ .
• Exponential integrals and error functions — mpmath v0.13 documentation 22 January 2010 19:32 UTC mpmath.googlecode.com [Source type: Academic]

^ And the first derivative and first differences of an exponential function are exponential.

In particular,
$\,{d \over dx} e^x = e^x.$
.That is, ex is its own derivative and hence is a simple example of a pfaffian function.^ An example of a function whose derivative is the same as the function.
• Calculus: Derivatives of Exponential Functions | MindBites.com 22 January 2010 19:32 UTC www.mindbites.com [Source type: Original source]

^ According to Wikipedia, the exponential function is the function ex, where e is the number such that the function ex equals its own derivative.
• How to get exponential success on your blog 22 January 2010 19:32 UTC www.catswhoblog.com [Source type: General]

^ Further, two function types have simple derivatives and related integration results (we shall learn about derivatives and integration in calculus).
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

.Functions of the form Kex for constant K are the only functions with that property.^ When we are integrating the Logarithmic and exponential functions we can multiply constants that have no effect on the derivative, they are there only to satisfy initial condition.

^ Exponential Density Function An exponential density function is a function of the form f(x) = ae ax (a a positive constant) with domain [0 + ).
• Calculus and Probability 2 22 January 2010 19:32 UTC people.hofstra.edu [Source type: Academic]

^ Any exponentially growing number will eventually grow larger than any other number which grows at only a constant rate for the same amount of time (and will also grow larger than any function which grows only subexponentially ).
• Exponential growth - encyclopedia article - Citizendium 2 February 2010 14:34 UTC en.citizendium.org [Source type: FILTERED WITH BAYES]

(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.^ Well, you say, what's the exponential function?
• Dr. Albert Bartlett: Arithmetic, Population and Energy (transcript) | Global Public Media 2 February 2010 14:34 UTC www.globalpublicmedia.com [Source type: FILTERED WITH BAYES]

^ Fact (Limits of exponential functions) y .
• Lesson 14: Exponential Functions 22 January 2010 19:32 UTC www.slideshare.net [Source type: Reference]

^ The equation of an exponential function is as follows: .
• Chapter 3: Fun with Functions 22 January 2010 19:32 UTC psych.fullerton.edu [Source type: Academic]

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.^ And the first derivative and first differences of an exponential function are exponential.

^ "The derivative of an exponential function with base a .
• Derivatives of exponential and logarithmic functions - An approach to calculus 2 February 2010 14:34 UTC www.themathpage.com [Source type: FILTERED WITH BAYES]

^ To reiterate: the derivative of an exponential function is a constant times that function.
• Lesson 16: Derivatives of Logarithmic and Exponential Functions 22 January 2010 19:32 UTC www.slideshare.net [Source type: Reference]

.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.^ We have considered exponential growth, logistic growth, and exponential decay.
• Analyzing Exponential and Logarithmic Functions - CK12 - Flexbooks 2 February 2010 14:34 UTC authors.ck12.org [Source type: Reference]

^ Graph B: Starting population of 100 at 1% exponential growth rate .
• Logistic Growth Versus Exponential Growth 2 February 2010 14:34 UTC members.optusnet.com.au [Source type: Academic]

^ Table E: Starting population of 100 at 2% exponential growth rate .
• Linear Growth Versus Exponential Growth 2 February 2010 14:34 UTC members.optusnet.com.au [Source type: FILTERED WITH BAYES]

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 ex

A continued fraction for ex can be obtained via an identity of Euler:
$\, \ e^x=1+\cfrac{x}{1-\cfrac{x}{x+2-\cfrac{2x}{x+3-\cfrac{3x}{x+4-\cfrac{4x}{x+5-\cfrac{5x}{x+6-\ddots}}}}}}$
The following generalized continued fraction for e2x/y converges more quickly; for x = y the first partial denominator will be zero but this does not stop convergence:
$e^\frac{2x}{y} = 1+\cfrac{2x}{y-x+\cfrac{x^2}{3y+\cfrac{x^2}{5y+\cfrac{x^2}{7y+\cfrac{x^2}{9y+\cfrac{x^2}{11y+\cfrac{x^2}{13y+\ddots\,}}}}}}}$

## Complex plane

.
Exponential function on the complex plane.
^ The function f(z) = z 2 is differentiable everywhere in the complex plane.

^ The complex logarithm is the inverse of the complex exponential function.

^ Suppose the complex function f(z) is differentiable on and within a closed curve C in the complex plane, and let R be the region inside the closed curve C .

.The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right.^ The graph of an exponential growth function increases when moving from left to right.

^ We show the the derivative of the exponential function is itself!
• Lesson 16: Derivatives of Logarithmic and Exponential Functions 22 January 2010 19:32 UTC www.slideshare.net [Source type: Reference]

^ The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable.
• How to get exponential success on your blog 22 January 2010 19:32 UTC www.catswhoblog.com [Source type: General]

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.^ Exponential decay: An Exponential Decay Function is a function of the form: .
• Exponential growth and decay made easy 2 February 2010 14:34 UTC classes.tmcc.edu [Source type: Academic]

^ A density function of this form is referred to as an exponential density function .
• Calculus and Probability 2 22 January 2010 19:32 UTC people.hofstra.edu [Source type: Academic]

^ On a functional equation for the exponential function of a complex variable.
• Complex Exponential Function 22 January 2010 19:32 UTC math.fullerton.edu [Source type: Academic]

.Some of these definitions mirror the formulas for the real-valued exponential function.^ Substituting these values in the formula gives (n = 7) .
• Regression: Fitting Functions to Data 22 January 2010 19:32 UTC people.hofstra.edu [Source type: FILTERED WITH BAYES]

^ Any real function can be approximated by a series > of exponentials.
• Processes which propagate faster than exponentially - sci.astro | Google Groups 22 January 2010 19:32 UTC groups.google.com [Source type: FILTERED WITH BAYES]

^ Any real function can be approximated by a series of exponentials.
• Processes which propagate faster than exponentially - sci.astro | Google Groups 22 January 2010 19:32 UTC groups.google.com [Source type: FILTERED WITH BAYES]

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.^ Using the formula above is easy, as the following example shows.
• Regression: Fitting Functions to Data 22 January 2010 19:32 UTC people.hofstra.edu [Source type: FILTERED WITH BAYES]

^ TabLists.com - To Do Lists, Task List, Powerful & Easy to Use Pitashi: Jeff Solomon's Guide to Everything; precisely complicated and exactly complex.
• The Complex System: Story of an Entrepreneur: 10 Keys to Exponential Growth 2 February 2010 14:34 UTC www.thecomplexsystem.com [Source type: General]

^ My statistical analysis below will show you why using a linear growth trend is often the right way to go.
• Consistent Cash Creators, Part 2: Linear vs. Exponential Growth | Fat Pitch Financials 2 February 2010 14:34 UTC www.fatpitchfinancials.com [Source type: FILTERED WITH BAYES]

.Another definition extends the real exponential function.^ Definition of Exponential Function The function f defined by .
• College Algebra Tutorial on Exponential Functions 22 January 2010 19:32 UTC www.wtamu.edu [Source type: General]

^ This website will help you with the definition of and graphing exponential functions.
• College Algebra Tutorial on Exponential Functions 22 January 2010 19:32 UTC www.wtamu.edu [Source type: General]

^ Note that expanse of exponential function is along x – axis on either side of the y-axis, showing that its domain is R. On the other hand, the expanse of “y” is limited to positive side of y-axis, showing that its range is positive real number.
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

First, we state the desired property ex + iy = exeiy. .For ex we use the real exponential function.^ Graphing exponential functions using transformations.
• Analyzing Exponential and Logarithmic Functions - CK12 - Flexbooks 2 February 2010 14:34 UTC authors.ck12.org [Source type: Reference]

^ Any real function can be approximated by a series > > of exponentials.
• Processes which propagate faster than exponentially - sci.astro | Google Groups 22 January 2010 19:32 UTC groups.google.com [Source type: FILTERED WITH BAYES]

^ The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable.
• How to get exponential success on your blog 22 January 2010 19:32 UTC www.catswhoblog.com [Source type: General]

We then proceed by defining only: eiy = cos(y) + isin(y). .Thus we use the real definition rather than ignore it.^ At a certain level of complexity, we would rather concentrate on reducing the cost per microprocessor and using multiple processors than to continue adding complexity to a single processor.
• Exponential Growth an Illusion?: Response to Ilkka Tuomi 2 February 2010 14:34 UTC www.kurzweilai.net [Source type: FILTERED WITH BAYES]

^ I’m just pointing out the defacto definition is different than what you very highly educated people are using, and the defacto definition makes sense.
• Exponential Growth in Physical Systems #2 « Climate Audit 2 February 2010 14:34 UTC climateaudit.org [Source type: FILTERED WITH BAYES]

^ It is cited when explaining transition to turbulence rather than real, honest to goodness turbulence.
• Exponential Growth in Physical Systems #3 « Climate Audit 2 February 2010 14:34 UTC climateaudit.org [Source type: FILTERED WITH BAYES]

[5]
When considered as a function defined on the complex plane, the exponential function retains the important properties
$\,\!\, e^{z + w} = e^z e^w$
$\,\!\, e^0 = 1$
$\,\!\, e^z e 0$
$\,\!\, {d \over dz} e^z = e^z$
for all z and w.
It is a holomorphic function which is periodic with imaginary period 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.^ The exponent of the exponential function is inverse trigonometric function.
• Domain and range of exponential and logarithmic function 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ The exponential integral is related to the hyperbolic and trigonometric integrals (see chi() , shi() , ci() , si() ) similarly to how the ordinary exponential function is related to the hyperbolic and trigonometric functions: .
• Exponential integrals and error functions — mpmath v0.13 documentation 22 January 2010 19:32 UTC mpmath.googlecode.com [Source type: Academic]

^ Domain and range of exponential and logarithmic function Inverse functions (exercise) Inverse trigonometric functions More similar content » Collections using this module .
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

.Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.^ Exponential functions have one of each.
• Exponential Functions 2 February 2010 14:34 UTC fym.la.asu.edu [Source type: FILTERED WITH BAYES]

^ Intuitively, one way to see the log is to “correct” for growth.
• An Intuitive Guide To Exponential Functions & e | BetterExplained 2 February 2010 14:34 UTC betterexplained.com [Source type: General]

^ Every time one paradigm ran out of steam, another paradigm came out of left field to continue the exponential growth.
• Ray Kurzweil on how technology will transform us | Video on TED.com 2 February 2010 14:34 UTC www.ted.com [Source type: FILTERED WITH BAYES]

.Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z).^ The nature of logarithmic function is dependent on the base value.
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Pingback by Demystifying the Natural Logarithm (ln) .
• An Intuitive Guide To Exponential Functions & e | BetterExplained 2 February 2010 14:34 UTC betterexplained.com [Source type: General]

^ Computes the logarithm of the magnitude of the gamma function and its sign, subject to x not being a negative integer, and returns them as multiple values.

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.^ Some parameters are more fixed than others because there is less uncertainty about their true values.
• Exponential Growth in Physical Systems #2 « Climate Audit 2 February 2010 14:34 UTC climateaudit.org [Source type: FILTERED WITH BAYES]

^ After reading the examples in this lesson, you should be able to write a function to represent a given situation, to evaluate the function for a given value of x , and to solve exponential equations in order to find values of x , given values of the function.
• Analyzing Exponential and Logarithmic Functions - CK12 - Flexbooks 2 February 2010 14:34 UTC authors.ck12.org [Source type: Reference]

^ We will also look at the doubling time for an increasing exponential (the time required for a given increasing exponential function to double is always the same, regardless of the starting time) and the half-life (the time required for a given decreasing exponential function to lose half its value is always the same regardless of the starting time) of an exponential function.
• Introduction to Exponential Functions 2 February 2010 14:34 UTC vhcc2.vhcc.edu [Source type: FILTERED WITH BAYES]

Because it is multi-valued the rule about multiplying exponents for positive real numbers doesn't work in general:
$\,\!\, (e^z)^w e e^\left(z w\right)$
.See failure of power and logarithm identities for more about problems with combining powers.^ More importantly, the problems that Gerry was talking about will prevent these from giving good aggregate statistics.
• Exponential Growth in Physical Systems #2 « Climate Audit 2 February 2010 14:34 UTC climateaudit.org [Source type: FILTERED WITH BAYES]

^ The diagram above illustrates the ranges of outliers and extremes in the "classic" box and whisker plot (for more information about box plots, see Tukey, 1977).
• Statistics Glossary: E 22 January 2010 19:32 UTC www.statsoft.com [Source type: Academic]

^ For example, the following diagram illustrates the ranges of outliers and extremes in the "classic" box and whisker plot (for more information about box plots, see Tukey, 1977).
• Statistics Glossary: E 22 January 2010 19:32 UTC www.statsoft.com [Source type: Academic]

.The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.^ Exponential and logarithmic functions .
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Logarithmic and exponential functions are closely related functions.
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ On a functional equation for the exponential function of a complex variable.
• Complex Exponential Function 22 January 2010 19:32 UTC math.fullerton.edu [Source type: Academic]

.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.^ So, it seems like a good guess is that this special number e will actually be smaller than three, but pretty close to three—closer to three than it is to two.
• Calculus: Derivatives of Exponential Functions | MindBites.com 22 January 2010 19:32 UTC www.mindbites.com [Source type: Original source]

^ Linear coordinates, used in the original publication, give an upwardly curving line that may appear, to the eye, two comprise two linear segments.
• Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research 2 February 2010 14:34 UTC www.ncbi.nlm.nih.gov [Source type: Academic]

^ Note that expanse of exponential function is along x – axis on either side of the y-axis, showing that its domain is R. On the other hand, the expanse of “y” is limited to positive side of y-axis, showing that its range is positive real number.
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

## Computation of ez for a complex z

If z = x + yi, 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 ab where both a and b are complex

Complex exponentiation ab can be defined by converting a to polar coordinates and using the identity (eln(a))b = ab:
$\,a^b = (re^{{ heta}i})^b = (e^{\ln(r) + { heta}i})^b = e^{(\ln(r) + { heta}i)b}.$
.However, when b is not an integer, this function is multivalued, because θ is not unique (see failure of power and logarithm identities).^ The base "a" can not be zero, because 0 x is not uniquely defined as required for invertible function.
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Logarithms are closely related to power functions, so you may find it helpful to first review this subject.

^ Computes the logarithm of the gamma function, log Γ ( x ) , subject to x not being a negative integer.

## Matrices and Banach algebras

.The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any Banach algebra B.^ The general forms of the exponential function include .
• Introduction to Exponential Functions 2 February 2010 14:34 UTC vhcc2.vhcc.edu [Source type: FILTERED WITH BAYES]

^ Therefore power functions in general are sometimes also referred to as exponential functions.

^ Any real function can be approximated by a series > of exponentials.
• Processes which propagate faster than exponentially - sci.astro | Google Groups 22 January 2010 19:32 UTC groups.google.com [Source type: FILTERED WITH BAYES]

In this setting, e0 = 1, and ex is invertible with inverse ex for any x in B. If xy =yx, then ex+y = exey, but this identity can fail for noncommuting x and y.
.Some alternative definitions lead to the same function.^ Sometimes experimental error leads to the finding that two measurements over some period of time are the same, suggesting that perhaps there was no growth during that period.
• Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell-cycle research 2 February 2010 14:34 UTC www.ncbi.nlm.nih.gov [Source type: Academic]

^ On Some Series Arising From a Definition of the Exponential Function    J. K. L. MacDonald; F. R. Sharpe   The American Mathematical Monthly, Vol.
• Complex Exponential Function 22 January 2010 19:32 UTC math.fullerton.edu [Source type: Academic]

For instance, ex can be defined as $\lim_{n o \infty} \left(1+\frac{x}{n} \right)^n$. Or ex can be defined as f(1), where fRB is the solution to the differential equation f′(t) = xf(t) with initial condition f(0) = 1.

## On Lie algebras

.Given a Lie group G and its associated Lie algebra $\mathfrak{g}$, the exponential map is a map $\mathfrak{g} o G$ satisfying similar properties.^ MathSciNet.   Methods for the approximation of the matrix exponential in a Lie-algebraic setting Celledoni E.; Iserles A. IMA Journal of Numerical Analysis, 1 April 2001, vol.
• The Matrix Exponential 22 January 2010 19:32 UTC ecs.fullerton.edu [Source type: Academic]

^ In other words, we can say that an exponential function associates every real number (x) to a function given by : .
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ Tags: exponential functions properties of exponents graphs of exponential functions How to find an equation of an exponential function when an algebraic relationship is given.
• Exponential Functions - Concept - Precalculus - Brightstorm 22 January 2010 19:32 UTC www.brightstorm.com [Source type: Reference]

.In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation.^ Real roots and exponentials for complex numbers .
• 10. ln(x) and exp(x), Properties of 22 January 2010 19:32 UTC whyslopes.com [Source type: Academic]

^ Here x 2 > 0 for all real numbers x.
• 10. ln(x) and exp(x), Properties of 22 January 2010 19:32 UTC whyslopes.com [Source type: Academic]

^ Now, we know that argument (input to function) of logarithmic function is a positive number.
• Domain and range of exponential and logarithmic function 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

.Similarly, since the Lie group GL(n,R) of invertible n × n matrices has as Lie algebra M(n,R), the space of all n × n matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.^ The exponential function is designed to be invertible.
• Exponential and logarithmic functions 22 January 2010 19:32 UTC cnx.org [Source type: FILTERED WITH BAYES]

^ This is the case with exponential functions.
• Analyzing Exponential and Logarithmic Functions - CK12 - Flexbooks 2 February 2010 14:34 UTC authors.ck12.org [Source type: Reference]

^ This is not the case for exponential functions because of the x in the exponent.
• Analyzing Exponential and Logarithmic Functions - CK12 - Flexbooks 2 February 2010 14:34 UTC authors.ck12.org [Source type: Reference]

The identity exp(x+y) = exp(x)exp(y) can fail for Lie algebra elements x and y that do not commute; the Baker-Campbell-Hausdorff formula supplies the necessary correction terms.

## 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) = aax; 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.^ Whenever I’ve run into this kind of growth, I’ve been far more interested in the fact that it grows faster than any polynomial than in the fact that it grows slower than exp(n).
• Shtetl-Optimized » Blog Archive » A not-quite-exponential dilemma 22 January 2010 19:32 UTC scottaaronson.com [Source type: General]

^ Are there problems which go like N^N? Is it substantially faster than exponential time?
• Shtetl-Optimized » Blog Archive » A not-quite-exponential dilemma 22 January 2010 19:32 UTC scottaaronson.com [Source type: General]

^ Every time the growing quantity doubles, it takes more than all you’d used in all the proceeding growth.
• Dr. Albert Bartlett: Arithmetic, Population and Energy (transcript) | Global Public Media 2 February 2010 14:34 UTC www.globalpublicmedia.com [Source type: FILTERED WITH BAYES]

.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.^ This is an exponential increase in which the population doubles each generation.

^ The first example is the exponential growth function y = 1 (3) x .
• 12.3 - Exponential functions 2 February 2010 14:34 UTC www.mathonweb.com [Source type: Academic]

^ For example the function y = (¼) t describes exponential decay.
• 12.3 - Exponential functions 2 February 2010 14:34 UTC www.mathonweb.com [Source type: Academic]

## Similar properties of e and the function ez

The function ez is not in C(z) (i.e., is not the quotient of two polynomials with complex coefficients).
For n distinct complex numbers {a1,..., an}, the set {ea1z,..., eanz} is linearly independent over C(z).
The function ez is transcendental over C(z).

## References

1. ^ Goldstein, Lay, Schneider, Asmar, Brief calculus and its applications, 11th ed., Prentice-Hall, 2006.
2. ^ "The natural exponential function is identical with its derivative. This is really the source of all the properties of the exponential function, and the basic reason for its importance in applications…" - p.448 of Courant and Robbins, What is mathematics? An elementary approach to ideas and methods (edited by Stewart), 2nd revised edition, Oxford Univ. Press, 1996.
3. ^ a b [1]
4. ^ Eli Maor, E: the history of a number, p.156.
5. ^ Ahlfors, Lars V. (1953). Complex analysis. McGraw-Hill Book Company, Inc..

# Quotes

Up to date as of January 14, 2010

### From Wikiquote

.Quotes regarding exponential growth and the exponential function.^ Review linear and exponential growth functions.
• Emilie Mason, Teacher Instructions, Growth 2 February 2010 14:34 UTC score.kings.k12.ca.us [Source type: Academic]

^ (Redirected from Exponential growth ) Jump to: navigation , search Quotes regarding exponential growth and the exponential function .
• Exponential function - Wikiquote 2 February 2010 14:34 UTC en.wikiquote.org [Source type: Reference]

^ The first example is the exponential growth function y = 1 (3) x .
• 12.3 - Exponential functions 2 February 2010 14:34 UTC www.mathonweb.com [Source type: Academic]

• The greatest shortcoming of the human race is our inability to understand the exponential function
Albert A. Bartlett, physicist
.
• Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist.^ IV. Exponential growth in a finite environment .
• Exponential growth in a finite environment - Forgotten Fundamentals of the Energy Crisis Part 4 - article by Al Bartlett 2 February 2010 14:34 UTC www.albartlett.org [Source type: FILTERED WITH BAYES]

^ "Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist" --Kenneth Boulding       .
• EcoFuture™ Population and Sustainability - Exponential Growth and The Rule of 70 2 February 2010 14:34 UTC www.ecofuture.org [Source type: FILTERED WITH BAYES]

^ Real world models for exponential growth .
• Exponential Growth. How the graph relates to the equation and formula. Practice problems 2 February 2010 14:34 UTC www.mathwarehouse.com [Source type: Academic]

w:Kenneth Boulding, economist

# Study guide

Up to date as of January 14, 2010

# Simple English

[[File:|right|thumb|Three different functions: Linear (red), Cubic (blue) and Exponential (green).]]

In mathematics, an exponential function is a function that quickly grows. More precisely, it is the function $exp\left(x\right) = e^x$, where e is Euler's constant, an irrational number approximating 2.71828. Because the number e can be formulated as an infinite limit $e = \lim_\left\{n \to \infty\right\} \left(1+ \frac\left\{1\right\}\left\{n\right\}\right)^n$, exp(x) is thus that formula to the xth power, or $exp\left(x\right) = e^x = \lim_\left\{n \to \infty\right\} \left(1 + \frac\left\{1\right\}\left\{n\right\}\right)^\left\{nx\right\}$

## Properties

Because and although exponential functions use exponentiation, they follow the same rules. Thus,

$exp\left(x+y\right) = exp\left(x\right)exp\left(y\right) = e^\left\{x+y\right\}$. This follows the rule that $x^a \cdot x^b = x^\left\{a+b\right\}$.

The natural logarithm is the inverse operation of an exponential function.

$ln\left(x\right) = log_\left\{e\right\}\left(x\right) = \frac\left\{log\left(x\right)\right\}\left\{log\left(e\right)\right\}$

This may also seem unintuitive because of a fundamental property in differential calculus:

$\frac\left\{d\right\}\left\{dx\right\} e^x = e^x$, not $xe^x$ as it would follow the general rule of thumb (see differentiation)

This means that the slope of an exponential function is an exponential function itself, and subsequently this means it has a slope of 1 at x=0. These properties are the reason it is an important function in mathematics.

## Applications

The exponential function is among the most useful of mathematical functions. It is used to represent exponential growth, which has uses in virtually all science subjects and also in Finance where it has prominence.

## Examples

One example of an exponential function in real life would be interest in a bank. If a person deposits £100 into an account which gets 3% interest a month then the balance each month would be (assuming the money is untouched):

Month Balance Month Balance
January £100.00 July £119.41
February £103.00 August £122.99
March £106.09 September £126.68
April £109.27 October £130.48
May £112.55 November £134.39
June £115.93 December £138.42

Notice how the extra money from interest increases each month. The greater the original balance, the more interest the person will get.

Two mathematical examples of exponential functions are shown below.

a=2

$x$ Result
-2 0.25
-1 0.5
0 1
1 2
2 4
3 8
4 16

a=3

$x$ Result
-2 1/9
-1 1/3
0 1
1 3
2 9
3 27
4 81

## Relation to the mathematical constant e

Even though the base ($a$) can be any number bigger than zero, for example, 10 or 1/2, often it is a special number called e. The number e cannot be written exactly, but it is almost equal to 2.71828.

The number e is important to every exponential function. For example, a bank pays interest of 0.01 percent every day. One person takes his interest money and puts it in a box. After 10,000 days (about 30 years), he has 2 times as much money as he started with. Another person takes his interest money and puts it back into the bank. Because the bank now pays him interest on his interest, the amount of money is an exponential function. After 10,000 days, he does not have 2 times as much money as he started with, but he has 2.718145 times as much money as he started with. This number is very close to the number e. If the bank pays interest more often, so the amount paid each time is less, then the number will be closer to the number e.

A person can also look at the picture to see why the number e is important for exponential functions. The picture has three different curves. The curve with the black points is an exponential function with a base a little smaller than e. The curve with the short black lines is an exponential function with a base a little bigger than e. The blue curve is an exponential function with a base exactly equal to e. The red line is a tangent to the blue curve. It touches the blue curve at one point without crossing it. A person can see that the red curve crosses the x-axis, the line that goes from left to right, at -1. This is true only for the blue curve. This is the reason that the exponential function with the base e is special.

# Citable sentences

Up to date as of December 19, 2010

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