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Topic: Definitions of the exponential function

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Exponential function

E (mathematical constant)

Proof that e is irrational

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	Power and logarithm formulas. Exponential function exp (x), number e. Decibel (dB). ClasCalc - the classic and formula ...
		Definition of the binary logarithm, the natural logarithm, the decimal logarithm; exponential functions exp (x), number e.
		Definition of the logarithm, the binary logarithm, the natural logarithm, the decimal logarithm.
		The inverse trigonometric functions — definitions and properties of the inverse trigonometric functions.
	www.clascalc.com /logarithm.htm (726 words)

	Exponential function Summary
		Scientists from various fields use exponential functions as models for growth and decay phenomena in which a quantity is assumed to grow or decay at a rate which is proportional to the amount of the quantity at any given time.
		Exponential functions and modifications of exponential functions are currently being used to study such phenomena as the growth of the internet, the spread of AIDS, the projected growth or decay of the national debt, and much more.
		The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
	www.bookrags.com /Exponential_function (1700 words)

	Function - LoveToKnow 1911
		The notion of functionality or functional relation of two magnitudes was thus of geometrical origin; but a function soon came to be regarded as an analytical expression, not necessarily an algebraic expression, containing the variable or variables.
		The definition of a continuous function (§ 9) admits of immediate extension; but it is very important to observe that a function of two or more variables may be a continuous function of each of the variables, when the rest are kept constant, without being a continuous function of its argument.
		The function may be given by specifying the domain of the argument and the rule of calculation, or else the function may have to be determined in accordance with certain conditions; for example, it may have to satisfy in a prescribed domain an assigned differential equation.
	www.1911encyclopedia.org /Function (13567 words)

	Exponential function - Wikipedia, the free encyclopedia
		The exponential function is one of the most important functions in mathematics.
		The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives.
		This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions.
	en.wikipedia.org /wiki/Exponential_function (1112 words)

	Exponential function - Medbib.com, the modern encyclopedia (Site not responding. Last check: 2007-10-28)
		The exponential function is nearly flat (climbing slowly) for negative values of x, climbs quickly for positive values of x, and equals 1 when x is equal to 0.
		The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
		The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential).
	www.medbib.com /Exponential_function (1080 words)

	5.3 Exponential Functions and Models
		Recall that a linear function g can be written as g(x) = mx + b where ___ represents the ____________ _________ in g(x) for each unit increase in ____.
		An exponential function is different from a linear function.
		Tell whether the exponential function is an increasing function or decreasing function.
	mtsu32.mtsu.edu:11197 /math_1710_5.3.htm (302 words)

	[No title]
		The combination of radial synaptic function and negative exponential activation function produces units that model a Gaussian (bell-shaped) function centered at the weight vector.
		Exponential function, with results normalized so that the sum of activations across the layer is 1.0.
		Functions that add new cases (i.e., rows of data) and/or variables (i.e., columns of data) to the end of the data set (the "bottom" or the right hand side, respectively).
	www.statsoft.com /textbook/glosa.html (2294 words)

	Computational Recursion and Mathematical Induction
		In the first, the value of the function is defined explicitly for n = 1; in the second, the value of the factorial function for any n (greater that 1) is defined in terms of the value of the same function for n ‑ 1.
		This is typical of all recursive definitions, where (to borrow Hofstadter’s phrase) a procedure is defined in terms of simpler versions of itself.
		Note, however, that this is not the “down-then-up” process generated by the computer as it interprets the procedure’s definition, but the process of the human learner’s mind as he or she is working on the problem.
	cse.proj.ac.il /recursion/Induction_Recursion.htm (2639 words)

	Complex exponential function
		The usual definition of w^z is in terms of exp.
		function is the entire function defined by exp(z) = e^z".
		definition seemed to imply that exp(z) is multi-valued.
	www.gatago.com /sci/math/3277461.html (2635 words)

	[No title]
		In the first, the value of the function is defined explicitly for n = 1; in the second, the value of the factorial function for any n (greater that 1) is defined in terms of the value of the same function for n 1.
		One reason for this difference in the “dynamic” interpretation of the definition is that for mathematicians, the object of interest is the entire function, not the computation of a particular value of it.
		The Tower-of-Hanoi Puzzle This well-known recursive procedure for transferring n rings from one pole to another can be viewed as either a definition (of an algorithm), or as part of proof-of-existence (of a solution or algorithm), or even as part of the proof that the transfer is possible in 2n — 1 moves.
	edu.technion.ac.il /Faculty/uril/Papers/FLM_Induction_&_Recursion.doc (2267 words)

	A REVIEW OF LOGARITHMS
		The exponential function f with base a is denoted by
		The function value will be positive because a positive base raised to any power is positive.
		For example if (4, 16) is a point on the graph of an exponential function, then (16, 4) would be the corresponding point on the graph of the inverse logarithmic function.
	www.sosmath.com /algebra/logs/log4/log4.html (290 words)

	Dictionary of Algorithms and Data Structures
		This is a dictionary of algorithms, algorithmic techniques, data structures, archetypal problems, and related definitions.
		Algorithms include common functions, such as Ackermann's function.
		We thank those who contributed definitions as well as many others who offered suggestions and corrections.
	www.nist.gov /dads (686 words)

	7 The exponential and related functions
		There are a number of equivalent definitions of the exponential function:
		As usual we obtain the graph of the inverse function by reflecting in the line y=x.
		Looking at the graph we see this is not 1-1 so we need to restrict the domain in order to define an inverse function.
	www.maths.soton.ac.uk /~rdi/ma155/lectures2/node2.html (154 words)

	PlanetMath: complex sine and cosine
		The periodicity of the functions causes that their inverse functions, the complex cyclometric functions, are infinitely multivalued; they can be expressed via the complex logarithm and square root (see general power) as
		The derivatives of sine function and cosine function are obtained either from the series forms or from (1):
		This is version 21 of complex sine and cosine, born on 2004-10-21, modified 2005-09-07.
	planetmath.org /encyclopedia/Sine3.html (296 words)

	7.3.9.1 FUNCTION COMMUNICATION
		Most variables should not be known outside their functions, lest a different part of the program change values of the variables to the detriment of the function.
		Function scope begins when a variable is declared in a function, and ends when the function ends.
		C++ functions can also be written in a way that they pass by reference but pass by value is the usual technique.
	www.aspire.cs.uah.edu /textbook/CPP7011.html (1015 words)

	Functions 2 - maths online Gallery
		is a puzzle type game in which a set of given functional expressions and graphs shall be associated with each other.
		Emphasis is laid on the relation between the geometrically intuitive form of the graphs are the corresponding numerical values.
		As in the previous applet, the functions are of the form
	www.univie.ac.at /future.media/moe/galerie/fun2/fun2.html (304 words)

	PlanetMath: definitions in trigonometry
		The above definitions are not fully rigorous, because we have not defined the word angle.
		are periodic functions on the real line, and have the same period.
		This is version 5 of definitions in trigonometry, born on 2003-08-30, modified 2005-06-15.
	planetmath.org /encyclopedia/DefinitionsInTrigonometry.html (164 words)

	Business Calculus at the Library of Math (Free Online Mathematics)
		Exponential and logarithmic functions are used throughout the sciences and are particular useful in business applications.
		Then limit theorems concerning limits of polynomials and rational functions state that at a point in the domain the limit is equal to the value of the function at that point.
		By using points on the graph of the function determined by a uniform width partition of the interval the upper boundary of the trapezoid is formed.
	libraryofmath.com /Business_Calculus.html (2071 words)

	Complex Exponential Function
		The exponential function and linearization of quadratic polynomials.
		The Exponential of Iteration of e^x - 1
		The exponential function characterized by an approximate functional equation.
	mathews.ecs.fullerton.edu /c2003/ExponentialFunBib/Links/ExponentialFunBib_lnk_3.html (936 words)

	Allwords.com Definition of exponential
		Denoting a function that varies according to the power of another quantity, ie a function in which the variable quantity is an exponent, eg if y = ax, then y varies exponentially with x.
		Denoting a logarithmic increase or decrease in numbers of a population, eg exponential growth of bacteria or an exponential decay of radioactive isotopes.
		The function ex, where e is a constant with a value of approximately 2.718.
	www.allwords.com /word-exponential.html (133 words)

	Analytical Engine Mathematical Function Library
		Function evaluation leaves the Mill in whatever operation state was last used by the function cards.
		With the exception of the square root, these functions are implemented from the most straightforward series expansion, and do not range reduce their arguments or exploit symmetry.
		The identities, like the functions upon which they are based, have the advantage of working regardless of the precision of the calculation.
	www.fourmilab.ch /babbage/library.html (1485 words)

	No Title
		There are three kinds of exponential graphs: increasing (a>1), constant (a=1), and decreasing (a<1).
		The exponential function is crucial for many areas of study, from business (where the stock market and your bank account grow exponentially), to biology (populations grow exponentially at times of their life cycles), to all other areas of science.
		The exponential function with e as its base is called the natural exponential function: it has the amazing property that the function is its own derivative and antiderivative.
	www.nku.edu /~longa/classes/2001spring/mat120/sections/7.2/7.2.html (310 words)

	exponential function - a Whatis.com definition
		When the exponent decreases by 1, the value of the function decreases by this same factor (it is divided by e).
		When the exponent increases by 1, the value of the base-10 function increases by a factor of 10; when the exponent decreases by 1, the value of the function becomes 1/10 as great.
		For a given, constant base such as e or 10, the exponential function "undoes" the logarithm function, and the logarithm undoes the exponential.
	whatis.techtarget.com /gDefinition/0,294236,sid44_gci213699,00.html (248 words)

	chapter VIA Exponentials and DE's
		The definitions of the exponential, logarithmic, and trigonometric functions are more advanced and sophisticated requiring an understanding of powers, roots, and geometry for similar triangles.
		Defining the natural exponential function: We could give a definition for the natural exponential function characterizing it solely as the solution to the differential equation y'=y where y(0)=1.
		Defining a function by declaring it to be the unique solution to a differential equation is not as unusual as it may seem.
	www.humboldt.edu /~mef2/book/ch6/VIA.htm (2856 words)

	lec20.html (Site not responding. Last check: 2007-10-28)
		This command defines the n-th term, with the exponent x^n for the exponential function.
		This is one of the definitions for the exponential function.
		The error between the actual series (and thus the exp() function) and the polynomial is of order x^10.
	www.math.uic.edu /~jan/mcs320/Lec20/lec20.html (388 words)

	Chapter VI.B Learning and Logarithms.
		This was not the original definition of a logarithm given by Napier in the late 16th century based on rates and the convenience of the decimal notation for fractions.
		Because the function L satisfies all the properties of other logarithmic functions, the function L is usually called the "natural logarithm function" and conventionally, when t > 0, L(t) is denoted ln(t).
		We continue to study this function more extensively in the next few sections, but for now we close our discussion with an example that applies of the basic properties of the natural logarithm to finding the derivatives of some complicated functions.
	www.humboldt.edu /~mef2/book/VIB.htm (2754 words)

	Exponential Functions (Site not responding. Last check: 2007-10-28)
		The graphs of various exponential functions are compared; in addition, a comparison with the graphs of polynomial functions is made.
		There is an exploration which looks at the approximation of the natural exponential function by polynomials.
		Use this LiveMath notebook to view an animation showing the graphs of a parametrized family of exponential functions.
	archives.math.utk.edu /visual.calculus/0/exp_log.5 (285 words)

	Characterizations of the exponential function - Wikipedia, the free encyclopedia
		As a special case of these considerations, we will see that the three most common definitions given for the mathematical constant e are also equivalent to each other.
		For instance, when the value of the function is defined by a sequence or series, the convergence of this sequence or series needs to be established.
		In this case, we define the natural logarithm function ln(x) first, and then define exp(x) as the inverse of the natural logarithm.
	en.wikipedia.org /wiki/Characterizations_of_the_exponential_function (673 words)

	2.2 Exponential, Hyper-exponential and Hypo-exponential Distributions (Site not responding. Last check: 2007-10-28)
		Given that X is a random variable, then we can define two important functions that can characterize the stochastic process: cumulative distribution function (cdf) and the probability density function (pdf).
		An important property of the exponential distribution is the memoryless property.
		From the definition of hypoexponential and hyperexponential distributions, one can see that these distributions consist of a number exponential ``phases'' or ``stages'' in series and parallel, respectively.
	www.cs.wm.edu /~riska/main/node10.html (476 words)

	Introduction to the exponential function (Site not responding. Last check: 2007-10-28)
		This is the crucial property that makes an exponential function different from any other function: it increases (or decreases) faster if its value is larger because its growth rate is directly proportional to its value.
		function of n (the step), since if we know n we can use the above formula to work out the result.
		This is called an exponential function of t, because the t is the exponent (which means the power).
	www.ucl.ac.uk /Mathematics/geomath/level1/expnb/MHexpnbintro.html (472 words)

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