Exponential function
The
exponential function is one of the most important
functions in
mathematics. It is written as exp(
x) or (where
e is the
base of the natural logarithm) and can be defined in two equivalent ways, the first an
infinite series, the second a
limit:
The graph of e^{x} does not ever touch the x axis, although it comes arbitrarily close.


Here stands for the
factorial of and can be any
real or
complex number, or even any element of a
Banach algebra or the field of
padic numbers.
If x is real, then exp(x) is positive and strictly increasing. Therefore its inverse function, the natural logarithm ln(x), is defined for all positive x. Using the natural logarithm, one can define more general exponential functions as follows:

for all and all real .
The exponential function also gives rise to the trigonometric functions (as can be seen from Euler's formula) and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.
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. Expressions involving fractions and roots can often be simplified using exponential notation because



The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own derivatives:

If a variable's growth or decay rate is proportional to its size, as is the case in unlimited population growth, continuously compounded interest or radioactive decay, then the variable can be written as a constant times an exponential function of time.
The exponential function thus solves the basic differential equation

and it is for this reason commonly encountered in differential equations. In particular the solution of linear ordinary
differential equations can frequently be written in terms of exponential functions. These equations include
Schrödinger equation and the
Laplace's equation as well as the equations for
simple harmonic motion.
When considered as a function defined on the complex plane, the exponential function retains the important properties




for all
z and
w. The exponential function on the complex plane is a
holomorphic function which is periodic with imaginary period which can be written as

where and are real values. This formula connects the exponential function with the
trigonometric functions, and this is the reason that extending the natural logarithm to complex arguments yields a multivalued function ln(
z). We can define a more general exponentiation:

for all complex numbers
z and
w.
This is then also a multivalued function. The above stated exponential laws remain true if interpreted properly as statements about multivalued functions.
It is easy to see, that the exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the centre at 0, noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.
The definition of the exponential function exp given above can be used verbatim for every Banach algebra, and in particular for square matrices. In this case we have

if (
we should add the general formula involving commutators here.)

 exp(x) is invertible with inverse exp(x)
 the derivative of exp at the point x is that linear map which sends u to exp(x)·u.
In the context of noncommutative 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 is a fixed element of the algebra and is any real number. This function has the important properties



Exponential map 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.
See also exponential growth.