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 notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) 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 opine that the exponential function is "the most important function in mathematics".
The exponential function satisfies the exponentiation identity
which, along with the definition , shows that for positive integers n, and relates the exponential function to the elementary notion of exponentiation. The base of the exponential function, its value at 1, , is a ubiquitous mathematical constant called Euler's number.
While other continuous nonzero functions that satisfy the exponentiation identity are also known as exponential functions, the exponential function exp is the unique real-valued function of a real variable whose derivative is itself and whose value at 0 is 1; that is, for all real x, and Thus, exp is sometimes called the natural exponential function to distinguish it from these other exponential functions, which are the functions of the form where the base b is a positive real number. The relation for positive b and real or complexx establishes a strong relationship between these functions, which explains this ambiguous terminology.
The real exponential function can also be defined as a power series. This power series definition is readily extended to complex arguments to allow the complex exponential function to be defined. The complex exponential 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 more abstract properties and characterizations of the exponential function, the exponential can be generalized to and defined for entirely different kinds of mathematical objects (for example, a square matrix or a Lie algebra).
The graph of is upward-sloping, and increases faster as x increases. 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 for distinguishing 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,
For b > 1, the function is increasing (as depicted for b = e and b = 2), because makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = 1/2); and for b = 1 the function is constant.
Euler's numbere = 2.71828... 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", or simply "the exponential function". Since any exponential function 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
The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is a complicated expression.
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 (in blue), and the sum of the first n + 1 terms of its power series (in red).
The real exponential function can be characterized in a variety of equivalent ways. It is commonly defined by the following power series:
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). The constant e can then 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
satisfying 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
By way of the binomial theorem and the power series definition, the exponential function can also be defined as the following limit:
It can be shown that every continuous, nonzero solution of the functional equation is an exponential function, with
The red curve is the exponential function. The black horizontal lines show where it crosses the green vertical lines.
now known as e. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.
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 the exponential function obeys the basic exponentiation identity,
which 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 derivative of the exponential function is equal to the value of the function. From any point P on the curve (blue), let a tangent line (red), and a vertical line (green) with height h be drawn, forming a right triangle with a base b on the x-axis. Since the slope of the red tangent line (the derivative) at 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, h must be equal to the ratio of h to b. Therefore, the base b must always be 1.
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,rate constant, or transformation constant.
Furthermore, for any differentiable function f, we find, by the chain rule:
This formula also converges, though more slowly, for z > 2. For example:
The exponential function e^z plotted in the complex plane from -2-2i to 2+2i
A complex plot of , with the argument represented by varying hue. The transition from dark to light colors shows that is increasing only to the right. The periodic horizontal bands corresponding to the same hue indicate that is periodic in the imaginary part of .
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:
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.
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.
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.
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.
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 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.
The function ez is not in C(z) (that is, is not the quotient of two polynomials with complex coefficients).
If a1, ..., an are distinct complex numbers, then ea1z, ..., eanz are linearly independent over C(z). It follows that ez is transcendental over C(z).
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) ...
^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…
^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.