Exponential function
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Nowadays the term exponential function is almost exclusively used as a shortcut for the natural exponential function e^{x}, where e is Euler's number, a 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 exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
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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.
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.
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Formal definition
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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:^{[4]}
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
It is also the following limit:^{[5]}
Overview
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^{[6]} to the number
now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.^{[6]}
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,
first given by Euler.^{[7]} 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,
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.
Derivatives and differential equations
The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative. In particular,
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:
Continued fractions for e^{x}
A continued fraction for e^{x} can be obtained via an identity of Euler:
The following generalized continued fraction for e^{z} converges more quickly:^{[8]}
or, by applying the substitution z = Template:Frac:
with a special case for z = 2:
This formula also converges, though more slowly, for z > 2. For example:
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Complex plane
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:
The exponential function is periodic with imaginary period and can be written as
where a and b are real values and on the right the real functions must be used if used as a definition^{[9]} (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
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; that is, 0 is a lacunary value of the exponential function. 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 multivalued function.
We can then define a more general exponentiation:
for all complex numbers z and w. This is also a multivalued function, even when z is real. This distinction is problematic, as the multivalued 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 multivalued context:
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.
Computation of a^{b} where both a and b are complex
{{#invoke:main|main}} Complex exponentiation a^{b} can be defined by converting a to polar coordinates and using the identity (e^{ln(a)})^{b} = a^{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
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, 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.
Some alternative definitions lead to the same function. 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.
Lie algebras
Given a Lie group G and its associated Lie algebra , the exponential map is a map 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
{{#invoke:main|main}} The term double exponential function can have two meanings:
- a function with two exponential terms, with different exponents such as e^{3x} − e^{4x−2}
- 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) = 10^{10}, f(2) = 10^{100} = googol, …, f(100) = googolplex.
Factorials grow faster than exponential functions, but slower than double-exponential functions. Fermat numbers, generated by and double Mersenne numbers generated by are examples of double exponential functions.
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).
The function e^{z} is transcendental over C(z).
See also
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- Approximating Natural exponents (log base e)
- Carlitz exponential, a characteristic p analogue
- Characterizations of the exponential function
- e (mathematical constant)
- Exponential decay
- Exponential field
- Exponential growth
- Exponentiation
- Half-exponential function – a compositional square root of an exponential function
- List of exponential topics
- List of integrals of exponential functions
- p-adic exponential function
- Padé approximation – it can be used to approximate the exponential function by a fraction of polynomial functions
- Tetration
References
- ↑ Goldstein, Lay, Schneider, Asmar, Brief calculus and its applications, 11th ed., Prentice–Hall, 2006.
- ↑ "This 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.
- ↑ "Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm)…" – p.12 of Converse and Durrell, Plane and spherical trigonometry, C.E. Merrill co., 1911.
- ↑ Walter Rudin, Real and Complex Analysis, McGraw-Hill, 3rd ed., 1986, ISBN 978-0-07-054234-1, page 1
- ↑ Eli Maor, e: the Story of a Number, p.156.
- ↑ ^{6.0} ^{6.1} Template:Cite web
- ↑ Eli Maor, e: the Story of a Number, p.156.
- ↑ "A.2.2 The exponential function." L. Lorentzen and H. Waadeland, Continued Fractions, Atlantis Studies in Mathematics, page 268.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
External links
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- Template:Planetmath reference
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- Derivative of exponential function interactive graph
- Weisstein, Eric W., "Exponential Function", MathWorld.
- Taylor Series Expansions of Exponential Functions at efunda.com
- Complex exponential interactive graphic
- Derivative of exp(x^{n}) by limit definition
- General exponential limit