This article is about the function f(x) = ex and its generalizations. For functions of the form f(x) = xr, see
Power function. For the bivariate function f(x,y) = xy, see
Exponentiation. For the representation of scientific numbers, see
E notation.
Exponential
The natural exponential function along part of the real axis
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]
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
The real exponential function can also be defined as a
power series, which is readily extended to complex arguments to define the complex exponential function . This function takes on all complex values except for 0 and is closely related to the complex
trigonometric functions, as shown 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
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.
Relation to more general exponential functions
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 numbere = 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 real 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, in fact, applicable to all complex numbers; see
§ Complex plane for the extension of to the complex plane. Using the power series, the constant e can be defined as
The term-by-term differentiation of this power series reveals that for all real x, leading to another common characterization of as the unique solution of the
differential equation
that satisfies the initial condition
Based on this characterization, the
chain rule shows that its inverse function, the
natural logarithm, satisfies for or This relationship leads to a less common definition of the real exponential function as the solution to the equation
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,
From any of these definitions it can be shown that e−x is the reciprocal of ex. For example from the differential equation definition, exe−x = 1 when x = 0 and its derivative using the
product rule is exe−x − exe−x = 0 for all x, so exe−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 ex 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 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 cex 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:
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.
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) = cekx 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:
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 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 zw 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:
(ez)w ≠ ezw, but rather (ez)w = e(z + 2niπ)w multivalued over integers n
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.
3D plots of real part, imaginary part, and modulus of the exponential function
z = Re(ex + iy)
z = Im(ex + iy)
z = |ex + iy|
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.
Graphs of the complex exponential function
Checker board key:
Projection onto the range complex plane (V/W). Compare to the next, perspective picture.
Projection into the , , and dimensions, producing a flared horn or funnel shape (envisioned as 2-D perspective image)
Projection into the , , and dimensions, producing a spiral shape ( range extended to ±2π, again as 2-D perspective image)
The second image shows how the domain complex plane is mapped into the range complex plane:
zero is mapped to 1
the real axis is mapped to the positive real axis
the imaginary axis is wrapped around the unit circle at a constant angular rate
values with negative real parts are mapped inside the unit circle
values with positive real parts are mapped outside of the unit circle
values with a constant real part are mapped to circles centered at zero
values with a constant imaginary part are mapped to rays extending from zero
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.
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 algebraB. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y.
Some alternative definitions lead to the same function. For instance, ex can be defined as
Or ex can be defined as fx(1), where fx : R → B is the solution to the differential equation dfx/dt(t) = xfx(t), with initial condition fx(0) = 1; it follows that fx(t) = etx for every t in R.
Lie algebras
Given a
Lie groupG 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.
Transcendency
The function ez is not in the rational function ring : it is not the quotient of two polynomials with complex coefficients.
If a1, ..., an are distinct complex numbers, then ea1z, ..., eanz are linearly independent over , and hence ez is
transcendental over .
Computation
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 ex − 1 directly, bypassing computation of ex. For example, if the exponential is computed by using its
Taylor series
^The notation ln x is the ISO standard and is prevalent in the natural sciences and secondary education (US). However, some mathematicians (for example,
Paul Halmos) have criticized this notation and prefer to use log x for the natural logarithm of x.
^In pure mathematics, the notation log x generally refers to the natural logarithm of x or a logarithm in general if the base is immaterial.
^Meier, John; Smith, Derek (2017-08-07). Exploring Mathematics. Cambridge University Press. p. 167.
ISBN978-1-107-12898-9.
^Converse, Henry Augustus; Durell, Fletcher (1911).
Plane and Spherical Trigonometry. Durell's mathematical series. C. E. Merrill Company. p.
12. Inverse Use of a Table of Logarithms; that is, given a logarithm, to find the number corresponding to it, (called its antilogarithm) ...[1]
^Courant; Robbins (1996). Stewart (ed.). What is Mathematics? An Elementary Approach to Ideas and Methods (2nd revised ed.).
Oxford University Press. p. 448.
ISBN978-0-13-191965-5. 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…
^
abO'Connor, John J.; Robertson, Edmund F. (September 2001).
"The number e". School of Mathematics and Statistics. University of St Andrews, Scotland. Retrieved 2011-06-13.
^Beebe, Nelson H. F. (2002-07-09).
"Computation of expm1 = exp(x)−1"(PDF). 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, University of Utah. Retrieved 2015-11-02.