In mathematics, a function in which the variable is in the exponent, eg 2x. The most important exponential function is y = ex which has the property that dy/dx = y for all values of x. So the rate of growth of ex is proportional to its size: the larger it is, the faster it grows - hence the popular usage of exponential to refer to any process of runaway growth. In its more general form y = Aekx, this function models many physical situations, such as laws of growth and decay, and the discharge of condensors. Radioactive substances obey an exponential decay law, given by y = Ae?kx, k > 0. The exponential series is 
. The trigonometric functions can also be defined in terms of exponentials, 
and Euler established the remarkable result: e? + 1 = 0.
The exponential function is one of the most important functions in mathematics.
As a function of the real variable x, the graph of y=e is always positive (above the x axis) and increasing (viewed left-to-right).
Sometimes, especially in the sciences, the term exponential function is reserved for functions of the form ka, where a, called the base, is any positive real number.
In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object;
Properties
Most simply, exponential functions multiply at a constant rate.
Using the natural logarithm, one can define more general exponential functions. The function
defined for all a >
Note that the equation above holds for a = e, since
Exponential functions "translate between addition and multiplication" as is expressed in the following exponential laws:
These are valid for all positive real numbers a and b and all real numbers x and y. Expressions involving fractions and roots can often be simplified using exponential notation because:
and, for any a > 0, real number b, and integer n > 1:
Derivatives and differential equations
The importance of exponential functions in mathematics and the sciences stems mainly from properties of their derivatives.
For exponential functions with other bases:
Thus any exponential function is a constant multiple of its own derivative.
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.
Furthermore for any differentiable function f(x), we find, by the chain rule:
.Formal definition
The exponential function e can be defined in a variety of equivalent ways, as an infinite series. In particular it may be defined by a power series:
or as the limit of a sequence:
In these definitions, n!
For further explanation of these definitions and a proof of their equivalence, see the article Definitions of the exponential function.
Numerical value
To obtain the numerical value of the exponential function, the infinite series can be rewritten as :
This expression will converge quickly if we can ensure that x is less than one.
On the complex plane
When considered as a function defined on the complex plane, the exponential function retains the important properties
for all z and w.
It is a holomorphic function which is periodic with imaginary period 2πi and can be written as
where a and b are real values.
Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z).
The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin.
Matrices and Banach algebras
The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices (in which case the function is called the matrix exponential).
In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:
where A is a fixed element of the algebra and t is any real number. This function has the important properties
On Lie algebras
The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M (n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
Double exponential function
The term double exponential function can have two meanings:
a function with two exponential terms, with different exponents a function ;
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