From Wikipedia, the free encyclopedia
The exponential function
y = e^{x}
.^ 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]} .^ 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 realworld phenomena using exponential functions. Investigation 5: Exploring Exponential Functions 22 January 2010 19:32 UTC www.cdli.ca [Source type: Academic]
.^ 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 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 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 xaxis 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 tangent to the curve at any point is always equal to its ycoordinate. 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]
.^ 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]
.^ 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 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]
^ See also pages that link to Exponential growth or to this page.
.^ 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








Overview
.^ 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
now known as
e.
.^ 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 online 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 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]
.^ 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,
first given by
Euler.
^{[4]} .^ We stated in the section on exponential functions , that exponential functions were onetoone. 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
e^{x}
.^ 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: .
.^ 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]
.^ 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]
.^ 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.
.^ 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]
.^ 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]
.^ On the zeros of the qanalogue 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).
.^ 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:
.^ 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, e^{x} is defined as the solution y to the equation
It is also the following limit:
The error term of this limitexpression is described by
where, the polynomial's degree (in x) in the term with denominator n^{k} is 2k.
Derivatives and differential equations
.^ 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 xaxis 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 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,
.^ 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]
.^ 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
PicardLindelöf 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.
.^ 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:
.^ 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]
.^ 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:
Continued fractions for e^{x}
The following
generalized continued fraction for e
^{2x/y} converges more quickly; for
x =
y the first partial denominator will be zero but this does not stop convergence:
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 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.
.^ 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]
.^ 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:
.^ 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]
.^ 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 yaxis, showing that its domain is R. On the other hand, the expanse of “y” is limited to positive side of yaxis, 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
e^{x + iy} = e^{x}e^{iy}.
.^ 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:
e^{iy} = cos(y) + isin(y).
.^ 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
for all z and w.
where
a and
b are real values.
.^ 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]
.^ 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]
.^ 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:
for all complex numbers
z and
w. This is also a multivalued function.
.^ 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 halflife (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 multivalued the rule about multiplying exponents for positive real numbers doesn't work in general:
.^ 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]
.^ 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]
.^ 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: Singlecell studies, cellculture studies, and the object of cellcycle 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 yaxis, showing that its domain is R. On the other hand, the expanse of “y” is limited to positive side of yaxis, 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]
Plots of the exponential function on the complex plane



Computation of e^{z} for a complex z
If z = x + yi, where x and y are real numbers, then
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}:
.^ 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 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,
e^{0} = 1, and
e^{x} is invertible with inverse
e^{−x} for any
x in
B. If
xy =
yx, then
e^{x+y} =
e^{x}e^{y}, but this identity can fail for noncommuting
x and
y.
.^ 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: Singlecell studies, cellculture studies, and the object of cellcycle 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,
e^{x} can be defined as
. Or
e^{x} can be defined as
f(1), where
f:
R→
B is the solution to the differential equation
f′(
t) =
xf(
t) with initial condition
f(0) = 1.
On Lie algebras
.^ MathSciNet. Methods for the approximation of the matrix exponential in a Liealgebraic 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]
.^ 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]
.^ 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
BakerCampbellHausdorff 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) = 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) = 10^{10}, f(2) = 10^{100} = googol, ..., f(100) = googolplex.
.^ 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). ShtetlOptimized » Blog Archive » A notquiteexponential 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? ShtetlOptimized » Blog Archive » A notquiteexponential 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]
.^ 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 e^{z}
The function e^{z} is not in C(z) (i.e., is not the quotient of two polynomials with complex coefficients).
For n distinct complex numbers {a_{1},..., a_{n}}, the set {e^{a1z},..., e^{anz}} is linearly independent over C(z).
See also
References
 ^ Goldstein, Lay, Schneider, Asmar, Brief calculus and its applications, 11th ed., PrenticeHall, 2006.
 ^ "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.
 ^ ^{a} ^{b} [1]
 ^ Eli Maor, E: the history of a number, p.156.
 ^ Ahlfors, Lars V. (1953). Complex analysis. McGrawHill Book Company, Inc..
External links