The natural exponential function Failed to parse (Missing texvc executable; please see math/README to configure.): y = e^x

In mathematics, the exponential function is the function e^x, where e is the number (approximately 2.718281828) such that the function e^x is its own derivative.^[1][2] The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The function is often written as exp(x), especially when it is impractical to write the independent variable as a superscript.

The graph of y = e^x is upward-sloping, and increases faster as x increases. 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 slope of the tangent to the graph at each point is equal to its y coordinate at that point. The inverse function is the natural logarithm ln(x); because of this, some old texts^[3] refer to the exponential function as the antilogarithm.

Sometimes the term exponential function is used more generally for functions of the form cb^x, where the base b is any positive real number, not necessarily e. See exponential growth for this usage.

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.

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 Defining e: proof that e is irrational  · representations of e · Lindemann–Weierstrass theorem People John Napier  · Leonhard Euler Schanuel's conjecture

Overview[link]

The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this that led Jacob Bernoulli in 1683^[4] to the number

Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^n,

now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.^[4]

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 instead interest is compounded daily, this becomes (1+x/365)³⁶⁵. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function,

Failed to parse (Missing texvc executable; please see math/README to configure.): \exp(x) = \lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n},

first given by Euler.^[5] This is one of a number of characterizations of the exponential function; others involve series or differential equations.

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity,

Failed to parse (Missing texvc executable; please see math/README to configure.): \exp(x+y) = \exp(x) \cdot \exp(y),

which is why it can be written as e^x.

The derivative (rate of change) of the exponential function is the exponential function itself. 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 function property leads to exponential growth and exponential decay.

The exponential function extends to an entire function on the complex plane. Euler's formula relates its values at purely imaginary arguments to trigonometric functions. 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.

Formal definition[link]

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 e^x can be characterized in a variety of equivalent ways. In particular it may be defined by the following power series^[6]:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

Less commonly, e^x is defined as the solution y to the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): x = \int_1^y {dt \over t}.

It is also the following limit:

Failed to parse (Missing texvc executable; please see math/README to configure.): e^x = \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^n.

Derivatives and differential equations[link]

The derivative of the exponential function is equal to the value of the function. From any point Failed to parse (Missing texvc executable; please see math/README to configure.): P

on the curve (blue), let a tangent line (red), and a vertical line (green) with height Failed to parse (Missing texvc executable; please see math/README to configure.): h
be drawn, forming a right triangle with a base Failed to parse (Missing texvc executable; please see math/README to configure.): b
on the Failed to parse (Missing texvc executable; please see math/README to configure.): x

-axis. Since the slope of the red tangent line (the derivative) at Failed to parse (Missing texvc executable; please see math/README to configure.): P

is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, Failed to parse (Missing texvc executable; please see math/README to configure.): h
must be equal to the ratio of Failed to parse (Missing texvc executable; please see math/README to configure.): h
to Failed to parse (Missing texvc executable; please see math/README to configure.): b

. Therefore the base Failed to parse (Missing texvc executable; please see math/README to configure.): b

must always be Failed to parse (Missing texvc executable; please see math/README to configure.): 1

.

The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative. In particular,

Failed to parse (Missing texvc executable; please see math/README to configure.): \,{d \over dx} e^x = e^x.

That is, e^x is its own derivative and hence is a simple example of a Pfaffian function. Functions of the form ce^x for constant c are the only functions with that property (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.

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. Explicitly for any real constant k, a function f: R→R satisfies f′ = kf if and only if f(x) = ce^kx for some constant c.

Furthermore for any differentiable function f(x), we find, by the chain rule:

Failed to parse (Missing texvc executable; please see math/README to configure.): {d \over dx} e^{f(x)} = f'(x)e^{f(x)}.

[edit] Continued fractions for e^x

A continued fraction for e^x can be obtained via an identity of Euler:

Failed to parse (Missing texvc executable; please see math/README to configure.): \, \ 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 e^2x/y converges more quickly:^{[citation needed]}

Failed to parse (Missing texvc executable; please see math/README to configure.): e^{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\,}}}}}}}

with a special case for x = y = 1:

Failed to parse (Missing texvc executable; please see math/README to configure.): e^2 = 7+\cfrac{2}{5+\cfrac{1}{7+\cfrac{1}{9+\cfrac{1}{11+\cfrac{1}{13+\ddots.}}}}}

Complex plane[link]

Exponential function on the complex plane. The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right. 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. One such definition parallels the power series definition for real numbers, where the real variable is replaced by a complex one:

Failed to parse (Missing texvc executable; please see math/README to configure.): e^z = \sum_{n = 0}^\infty\frac{z^n}{n!}

The exponential function is periodic with imaginary period Failed to parse (Missing texvc executable; please see math/README to configure.): 2 \pi i

and can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): e^{a + bi} = e^a (\cos b + i \sin b)\,

where a and b are real values and on the right the real functions must be used if used as a definition^[7] (see also Euler's formula). This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions.

When considered as a function defined on the complex plane, the exponential function retains the properties

• Failed to parse (Missing texvc executable; please see math/README to configure.): e^{z + w} = e^z e^w\,

• Failed to parse (Missing texvc executable; please see math/README to configure.): e^0 = 1\,

• Failed to parse (Missing texvc executable; please see math/README to configure.): e^z \ne 0

• Failed to parse (Missing texvc executable; please see math/README to configure.): {d \over dz} e^z = e^z

• Failed to parse (Missing texvc executable; please see math/README to configure.): \,(e^z)^n = e^{nz}, n \in \mathbb{Z}

for all z and w.

The exponential function is an entire function as it is holomorphic over the whole complex plane. It takes on every complex number excepting 0 as value. This is an example of Picard's little theorem that any non-constant entire function takes on every complex number as value with at most one value excepted.

Extending the natural logarithm to complex arguments yields the complex logarithm log z, which is a multi-valued function.

We can then define a more general exponentiation:

Failed to parse (Missing texvc executable; please see math/README to configure.): z^w = e^{w \log z}\,

for all complex numbers z and w. This is also a multi-valued function, even when z is real. This distinction is problematic, as the multi-valued functions log z and z^w are easily confused with their single-valued equivalents when substituting a real number for z. The rule about multiplying exponents for the case of positive real numbers must be modified in a multi-valued context:

Failed to parse (Missing texvc executable; please see math/README to configure.): (e^z)^w \ne e^{z w}

, but rather Failed to parse (Missing texvc executable; please see math/README to configure.): (e^z)^w = e^{(z + 2\pi i n) w}\,

multivalued over integers n

See failure of power and logarithm identities for more about problems with combining powers.

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases might be noted: when the original line is parallel to the real axis, the resulting 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.

• Plots of the exponential function on the complex plane
• 

  z = Re(e^x+iy)

• 

  z = Im(e^x+iy)

• 

  z = |e^x+iy|

[edit] Computation of a^b where both a and b are complex

Complex exponentiation a^b can be defined by converting a to polar coordinates and using the identity (e^ln(a))^b = a^b:

Failed to parse (Missing texvc executable; please see math/README to configure.): \,a^b = (re^{{\theta}i})^b = (e^{\ln(r) + {\theta}i})^b = e^{(\ln(r) + {\theta}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).

Matrices and Banach algebras[link]

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. In this setting, e⁰ = 1, and e^x is invertible with inverse e^−x for any x in B. If xy =yx, then e^x+y = e^xe^y, but this identity can fail for noncommuting x and y.

Some alternative definitions lead to the same function. For instance, e^x can be defined as Failed to parse (Missing texvc executable; please see math/README to configure.): \textstyle \lim_{n \to \infty} \left(1+\frac{x}{n} \right)^n . 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[link]

Given a Lie group G and its associated Lie algebra Failed to parse (Missing texvc executable; please see math/README to configure.): \mathfrak{g} , the exponential map is a map Failed to parse (Missing texvc executable; please see math/README to configure.): \mathfrak{g}\to G

satisfying similar properties. 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. 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 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[link]

The term double exponential function can have two meanings:

• a function with two exponential terms, with different exponents^{[clarification needed]}
• a function f(x) = a^ax; 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. Fermat numbers, generated by Failed to parse (Missing texvc executable; please see math/README to configure.): \,F(m) = 2^{2^m} + 1

and double Mersenne numbers generated by Failed to parse (Missing texvc executable; please see math/README to configure.): \,MM(p) = 2^{(2^p-1)}-1
are examples of double exponential functions.

[edit] 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 {a₁,..., a_n}, the set {e^a₁z,..., e^a_nz} is linearly independent over C(z).

The function e^z is transcendental over C(z).

See also[link]

Mathematics portal

• e (mathematical constant)
• Exponential decay
• Characterizations of the exponential function
• Exponential field
• Exponential growth
• Exponentiation
• List of exponential topics
• List of integrals of exponential functions
• p-adic exponential function
• Carlitz exponential - a characteristic p analogue
• Tetration
• Padé approximation can be used to approximate the exponential function by a fraction of polynomial functions.

References[link]

External links[link]

• Complex exponential function, PlanetMath.org.
• Derivative of exponential function, PlanetMath.org.
• Derivative of exponential function interactive graph
• Weisstein, Eric W., "Exponential Function" from MathWorld.
• Taylor Series Expansions of Exponential Functions at efunda.com
• Complex exponential interactive graphic
• A relatively rigorous development of the exponential function

Albert Allen Bartlett (born 1923 in Shanghai)^[1] is an emeritus Professor of Physics at the University of Colorado at Boulder, USA. As of July 2001^[update] Professor Bartlett had lectured over 1,600 times since September, 1969 on Arithmetic, Population, and Energy.^[2][3] Bartlett regards the word combination "sustainable growth" as an oxymoron, since even modest annual percentage population increases will inevitably equate to huge exponential growth over sustained periods of time. He therefore regards human overpopulation as "The Greatest Challenge" facing humanity.

Career[link]

Bartlett joined the faculty of the University of Colorado in Boulder in September 1950. He has a B.A. degree in physics from Colgate University (1944), and M.A. (1948) and Ph.D. (1951) degrees in physics from Harvard University. In 1978 he was national president of the American Association of Physics Teachers. He is a Fellow of the American Physical Society and of the American Association for the Advancement of Science. In 1969 and 1970 he served two terms as the elected Chair of the four-campus Faculty Council of the University of Colorado.

Views on population growth[link]

Thomas Malthus, pioneer of the study of demographics.

Human population curve showing the power of exponential growth

Chart showing change in oil prices since the 19th century. The top curve is inflation-adjusted

World population from 1800 to 2100, based on UN 2004 projections (red, orange, green) and US Census Bureau historical estimates (black).

Professor Bartlett often explains how sustainable growth is a contradiction. His view is based on the fact that a modest percentage growth can equate to huge escalations over relatively short periods of time.^[4]

Bartlett argues that, over time, compound growth can yield enormous increases. For example, an investor earning a constant annual 7% return on their investment would find their capital doubling within 10 years. But the same exponential power, so advantageous to patient investors, may be potentially calamitous when applied to human population. A population of 10,000 individuals, if it were to grow at a constant rate of 7% per annum, would reach a population size of 10 million after 100 years.^[5]

Bartlett regards overpopulation as "The Greatest Challenge" facing humanity, and promotes sustainable living. He opposes the cornucopian school of thought (as advocated by people such as Julian Lincoln Simon), and refers to it as "The New Flat Earth Society"^[6]

J. B. Calvert (1999) has proposed that Bartlett's law ^[7] will result in the exhaustion of petrochemical resources due to the exponential growth of the world population (in line with the Malthusian Growth Model).

Bartlett has made two notable statements relating to sustainability:

"The greatest shortcoming of the human race is our inability to understand the exponential function."

and his Great Challenge:

"Can you think of any problem in any area of human endeavor on any scale, from microscopic to global, whose long-term solution is in any demonstrable way aided, assisted, or advanced by further increases in population, locally, nationally, or globally?"

Books[link]

• The Essential Exponential For the Future of Our Planet a collection of essays by Professor Bartlett (2004). Center for Science, Mathematics and Computer Education, University of Nebraska-Lincoln. ISBN 0-9758973-0-6 ^[8]

See also[link]

• Exponential growth
• Thomas Robert Malthus - the originator of the Malthusian catastrophe argument
• Peak oil
• M. King Hubbert, author of the Hubbert Curve.

Notes[link]

References[link]

• Professor talks at an exponential rate Energy Bulletin article by Todd Neff. Retrieved July 2011

External links[link]

• Professor Bartlett's website contains background, articles, book "The Essential Exponential", and links to his talk, "Arithmetic, Population, and Energy". Retrieved July 2011
• Professor Bartlett's page at Colorado University Retrieved July 2011
• Biography of Dr Bartlett Retrieved July 2011
• "Arithmetic, Population, and Energy" by Professor Albert Bartlett Free audio and video 57 minute lecture. Retrieved July 2011
• "Arithmetic, Population, and Energy" by Professor Albert Bartlett Text of lecture (extracts). Retrieved July 2011
• Analysis of Bartlett's "Arithmetic, Population, and Energy" presentation - Exponentialist website. Retrieved July 2011
• Is There a Population Problem? Ecofuture website. Retrieved July 2011
• The Massive Movement to Marginalise the Modern Malthusian Message article by Professor Bartlett. Retrieved July 2011
• Thoughts on Long-term Energy Supplies - Scientists and the Silent Lie article on energy and population in Physics Today (2004). Retrieved July 2011
• Clean Coal Technology Speech by Andrew McNamara MP on clean coal and report by Professor Bartlett
• Arithmetic, Population & Energy, Part I @YouTube.com (Part 1 of 8), (parts 2-4 are linked). Retrieved July 2011
• Arithmetic, Population & Energy, Part II @YouTube.com (Part 5 of 8), (parts 6-8 are linked). Retrieved July 2011
• Arithmetic, Population & Energy @Google (entire video). Retrieved July 2011
• Lecture by Professor Bartlett about the exponential function @Vimeo. Retrieved July 2011
• BLIND SPOT, Documentary, Population & Energy. Retrieved July 2011

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