The exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number (repeated multiplication), but various modern definitions allow it to be rigorously extended to all real arguments , including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".^{[1]}
Exponential | |
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General information | |
General definition | |
Domain, codomain and image | |
Domain | |
Image | |
Specific values | |
At zero | 1 |
Value at 1 | e |
Specific features | |
Fixed point | −W_{n}(−1) for |
Related functions | |
Reciprocal | |
Inverse | Natural logarithm, Complex logarithm |
Derivative | |
Antiderivative | |
Series definition | |
Taylor series |
The functions for positive real numbers are also known as exponential functions, and satisfy the exponentiation identity:This implies (with factors) for positive integers , where , relating exponential functions to the elementary notion of exponentiation. The natural base is a ubiquitous mathematical constant called Euler's number. To distinguish it, is called the exponential function or the natural exponential function: it is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1:
for all , and
The relation for and real or complex allows general exponential functions to be expressed in terms of the natural exponential.
More generally, especially in applied settings, any function defined by
is also known as an exponential function, as it solves the initial value problem , meaning its rate of change at each point is proportional to the value of the function at that point. This behavior models diverse phenomena in the biological, physical, and social sciences, for example the unconstrained growth of a self-reproducing population, the decay of a radioactive element, the compound interest accruing on a financial fund, or a growing body of manufacturing expertise.
The exponential function can also be defined as a power series, which is readily applied to real numbers, complex numbers, and even matrices. The complex exponential function takes on all complex values except for 0 and is closely related to the complex trigonometric functions by Euler's formula:
Motivated by its more abstract properties and characterizations, the exponential function can be generalized to much larger contexts such as square matrices and Lie groups. Even further, the differential equation definition can be generalized to a Riemannian manifold.
The exponential function for real numbers is a bijection from to the interval .^{[2]} Its inverse function is the natural logarithm, denoted ,^{[nb 1]} ,^{[nb 2]} or . Some old texts^{[3]} refer to the exponential function as the antilogarithm.
The graph of is upward-sloping, and increases faster as x increases.^{[4]} The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point.
The exponential function is sometimes called the natural exponential function in order to distinguish it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive b,
As functions of a real variable, exponential functions are uniquely characterized by the fact that the derivative of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:
For b > 1, the function is increasing (as depicted for b = e and b = 2), because makes the derivative always positive; this is often referred to as exponential growth. For positive b < 1, the function is decreasing (as depicted for b = 1/2); this is often referred to as exponential decay. For b = 1, the function is constant.
Euler's number e = 2.71828...^{[5]} is the unique base for which the constant of proportionality is 1, since , so that the function is its own derivative:
This function, also denoted as exp x, is called the "natural exponential function",^{[6]}^{[7]} or simply "the exponential function". Since any exponential function defined by can be written in terms of the natural exponential as , it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by or
The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is more complicated and harder to read in a small font.
For real numbers c and d, a function of the form is also an exponential function, since it can be rewritten as
The exponential function can be characterized in a variety of equivalent ways. It is commonly defined by the following power series:^{[1]}^{[8]}
Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers; see § Complex plane.
The term-by-term differentiation of this power series reveals that for all x, leading to another common characterization of as the unique solution of the differential equation that satisfies the initial condition
Solving the ordinary differential equation with the initial condition using Euler's method gives another common characterization, the product limit formula:^{[9]}^{[8]}
With any of these equivalent definitions, one defines Euler's number . It can then be shown that the exponentiation and the exponential function are equivalent.
It can be shown that every continuous, nonzero solution of the functional equation for is an exponential function in the more general sense, with
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 observation that led Jacob Bernoulli in 1683^{[10]} to the number now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.^{[10]}
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 Leonhard Euler.^{[9]} 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 e^{−x} is the reciprocal of e^{x}. For example from the differential equation definition, e^{x} e^{−x} = 1 when x = 0 and its derivative using the product rule is e^{x} e^{−x} − e^{x} e^{−x} = 0 for all x, so e^{x} e^{−x} = 1 for all x.
From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity. For example from the power series definition, This justifies the notation e^{x} for exp 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 or 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.
The importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and is equal to 1 when x = 0. That is,
Functions of the form ce^{x} for constant c are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Other ways of saying the same thing include:
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. The constant k is called the decay constant, disintegration constant,^{[11]} rate constant,^{[12]} or transformation constant.^{[13]}
Furthermore, for any differentiable function f, we find, by the chain rule:
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:^{[14]}
or, by applying the substitution z = x/y: with a special case for z = 2:
This formula also converges, though more slowly, for z > 2. For example:
As in the real case, the exponential function can be defined on the complex plane in several equivalent forms.
The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one:
Alternatively, the complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one:
For the power series definition, term-wise multiplication of two copies of this power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments:
The definition of the complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments.
In particular, when z = it (t real), the series definition yields the expansion
In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of cos t and sin t, respectively.
This correspondence provides motivation for defining cosine and sine for all complex arguments in terms of and the equivalent power series:^{[15]}
for all
The functions exp, cos, and sin so defined have infinite radii of convergence by the ratio test and are therefore entire functions (that is, holomorphic on ). The range of the exponential function is , while the ranges of the complex sine and cosine functions are both in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of , or excluding one lacunary value.
These definitions for the exponential and trigonometric functions lead trivially to Euler's formula:
We could alternatively define the complex exponential function based on this relationship. If z = x + iy, where x and y are both real, then we could define its exponential as where exp, cos, and sin on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.^{[16]}
For , the relationship holds, so that for real and maps the real line (mod 2π) to the unit circle in the complex plane. Moreover, going from to , the curve defined by traces a segment of the unit circle of length starting from z = 1 in the complex plane and going counterclockwise. Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions.
The complex exponential function is periodic with period 2πi and holds for all .
When its domain is extended from the real line to the complex plane, the exponential function retains the following properties:
for all
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 exist: 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.
Considering the complex exponential function as a function involving four real variables: the graph of the exponential function is a two-dimensional surface curving through four dimensions.
Starting with a color-coded portion of the domain, the following are depictions of the graph as variously projected into two or three dimensions.
The second image shows how the domain complex plane is mapped into the range complex plane:
The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image.
The third image shows the graph extended along the real axis. It shows the graph is a surface of revolution about the axis of the graph of the real exponential function, producing a horn or funnel shape.
The fourth image shows the graph extended along the imaginary axis. It shows that the graph's surface for positive and negative values doesn't really meet along the negative real axis, but instead forms a spiral surface about the axis. Because its values have been extended to ±2π, this image also better depicts the 2π periodicity in the imaginary value.
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 Exponentiation § Failure of power and logarithm identities).
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 unital 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_{x}(1), where f_{x} : R → B is the solution to the differential equation df_{x}/dt(t) = x f_{x}(t), with initial condition f_{x}(0) = 1; it follows that f_{x}(t) = e^{tx} for every t in R.
Given a Lie group G and its associated Lie algebra , the exponential map is a map ↦ 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 can fail for Lie algebra elements x and y that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.
The function e^{z} is not in the rational function ring : it is not the quotient of two polynomials with complex coefficients.
If a_{1}, ..., a_{n} are distinct complex numbers, then e^{a1z}, ..., e^{anz} are linearly independent over , and hence e^{z} is transcendental over .
When computing (an approximation of) the exponential function near the argument 0, the result will be close to 1, and computing the value of the difference with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large calculation error, possibly even a meaningless result.
Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1
, for computing e^{x} − 1 directly, bypassing computation of e^{x}. For example, if the exponential is computed by using its Taylor series
one may use the Taylor series of :
This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators,^{[17]}^{[18]} operating systems (for example Berkeley UNIX 4.3BSD^{[19]}), computer algebra systems, and programming languages (for example C99).^{[20]}
In addition to base e, the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: and .
A similar approach has been used for the logarithm (see lnp1).^{[nb 3]}
An identity in terms of the hyperbolic tangent, gives a high-precision value for small values of x on systems that do not implement expm1(x).
Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) ...[1]
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…
Berkeley UNIX 4.3BSD introduced the expm1() function in 1987.