Exponential function

In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted or ; the latter is preferred when the argument is a complicated expression. It is called exponential because its argument can be seen as an exponent to which a constant number, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.
The exponential function converts sums to products:. Its inverse function, the natural logarithm, or, converts products to sums:.
The exponential function is occasionally called the natural exponential function, matching the name natural logarithm, for distinguishing it from some other functions that are also commonly called exponential functions. These functions include the functions of the form, which is exponentiation with a fixed base. More generally, and especially in applications, functions of the general form are also called exponential functions. They grow or decay exponentially in that the rate that changes when is increased is proportional to the current value of.
The exponential function can be generalized to accept complex numbers as arguments. This reveals relations between multiplication of complex numbers, rotations in the complex plane, and trigonometry. Euler's formula expresses and summarizes these relations.
The exponential function can be even further generalized to accept other types of arguments, such as matrices and elements of Lie algebras.

Graph

The graph of is upward-sloping, and increases faster than every power of. The graph always lies above the -axis, but becomes arbitrarily close to it for large negative ; thus, the -axis is a horizontal asymptote. The equation means that the slope of the tangent to the graph at each point is equal to its height at that point.

Definitions and fundamental properties

There are several equivalent definitions of the exponential function, although of very different nature.

Differential equation

The exponential function is the unique differentiable function that equals its derivative, and takes the value for the value of its variable.
This definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.

Inverse of natural logarithm

The exponential function is the inverse function of the natural logarithm. That is,
for every real number and every positive real number

Power series

The exponential function is the sum of the power series
Image:Exp series.gif|right|thumb|The exponential function, and the sum of the first terms of its power series
where is the factorial of . This series is absolutely convergent for every, by the ratio test. This shows that the exponential function is defined for every, and is everywhere the sum of its Maclaurin series.

Functional equation

The exponential satisfies the functional equation
and maps the additive identity to the multiplicative identity.
The same equation is satisfied by other continuous functions that exponentiate their argument with an arbitrary base. Among these functions, the exponential function is characterized by the property that its derivative at is.

Limit of integer powers

The exponential function is the limit, as the integer goes to infinity,

Properties

Reciprocal: The functional equation implies. Therefore for every and
Positiveness: for every real number. This results from the intermediate value theorem, since and, if one would have for some, there would be an such that between and. Since the exponential function equals its derivative, this implies that the exponential function is monotonically increasing.
Extension of exponentiation to positive real bases: Let be a positive real number. The exponential function and the natural logarithm being the inverse each of the other, one has If is an integer, the functional equation of the logarithm implies
Since the right-most expression is defined if is any real number, this allows defining for every positive real number and every real number :
In particular, if is the Euler's number one has and thus This shows the equivalence of the two notations for the exponential function.

General exponential functions

A function is commonly called an exponential functionwith an indefinite articleif it has the form, that is, if it is obtained from exponentiation by fixing the base and letting the exponent vary.
More generally and especially in applied contexts, the term exponential function is commonly used for functions of the form. This may be motivated by the fact that, if the values of the function represent quantities, a change of measurement unit changes the value of, and so, it is nonsensical to impose.
These most general exponential functions are the differentiable functions that satisfy the following equivalent characterizations.

for every and some constants and.
for every and some constants and.
The value of is independent of.
For every the value of is independent of that is, for every,.

Image:Exponenciala priklad.png|thumb|200px|right|Exponential functions with bases 2 and 1/2
The base of an exponential function is the base of the exponentiation that appears in it when written as, namely. The base is in the second characterization, in the third one, and in the last one.

In applications

The last characterization is important in empirical sciences, as allowing a direct experimental test whether a function is an exponential function.
Exponential growth or exponential decaywhere the variable change is proportional to the variable valueare thus modeled with exponential functions. Examples are unlimited population growth leading to Malthusian catastrophe, continuously compounded interest, and radioactive decay.
If the modeling function has the form or, equivalently, is a solution of the differential equation, the constant is called, depending on the context, the decay constant, disintegration constant, rate constant, or transformation constant.

Equivalence proof

For proving the equivalence of the above properties, one can proceed as follows.
The two first characterizations are equivalent, since, if and, one has
The basic properties of the exponential function implies immediately the third and the last condition.
Suppose that the third condition is verified, and let be the constant value of Since the quotient rule for derivation
implies that
and thus that there is a constant such that
If the last condition is verified, let which is independent of. Using, one gets
Taking the limit when tends to zero, one gets that the third condition is verified with. It follows therefore that for some and As a byproduct, one gets that
is independent of both and.

Compound interest

The earliest occurrence of the exponential function was in Jacob Bernoulli's study of compound interests in 1683.
This is this study that led Bernoulli to consider the number
now known as Euler's number and denoted.
The exponential function is involved as follows in the computation of continuously compounded interests.
If a principal amount of 1 earns interest at an annual rate of compounded monthly, then the interest earned each month is times the current value, so each month the total value is multiplied by, and the value at the end of the year is. If instead interest is compounded daily, this becomes. 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.

Differential equations

Exponential functions occur very often in solutions of differential equations.
The exponential functions can be defined as solutions of differential equations. Indeed, the exponential function is a solution of the simplest possible differential equation, namely. Every other exponential function, of the form, is a solution of the differential equation, and every solution of this differential equation has this form.
The solutions of an equation of the form
involve exponential functions in a more sophisticated way, since they have the form
where is an arbitrary constant and the integral denotes any antiderivative of its argument.
More generally, the solutions of every linear differential equation with constant coefficients can be expressed in terms of exponential functions and, when they are not homogeneous, antiderivatives. This holds true also for systems of linear differential equations with constant coefficients.

Complex exponential

The exponential function can be naturally extended to a complex function, which is a function with the complex numbers as domain and codomain, such that its restriction to the reals is the above-defined exponential function, called real exponential function in what follows. This function is also called the exponential function, and also denoted or. For distinguishing the complex case from the real one, the extended function is also called complex exponential function or simply complex exponential.
Most of the definitions of the exponential function can be used verbatim for definiting the complex exponential function, and the proof of their equivalence is the same as in the real case.
The complex exponential function can be defined in several equivalent ways that are the same as in the real case.
The complex exponential is the unique complex function that equals its complex derivative and takes the value for the argument :
The complex exponential function is the sum of the series
This series is absolutely convergent for every complex number. So, the complex exponential is an entire function.
The complex exponential function is the limit
As with the real exponential function, the complex exponential satisfies the functional equation
Among complex functions, it is the unique solution which is holomorphic at the point and takes the derivative there.
The complex logarithm is a right-inverse function of the complex exponential:
However, since the complex logarithm is a multivalued function, one has
and it is difficult to define the complex exponential from the complex logarithm. On the opposite, this is the complex logarithm that is often defined from the complex exponential.
The complex exponential has the following properties:
and
It is periodic function of period ; that is
This results from Euler's identity and the functional identity.
The complex conjugate of the complex exponential is
Its modulus is
where denotes the real part of.