# Exponential function

The**exponential function**is one of the most important functionss in mathematics. It is written as exp(

*x*) or

*e*(where

^{x}*e*is the base of the natural logarithm) and can be defined in either of two equivalent ways, the first an infinite series, the second a limit of a sequence:

The graph of

*e*does

^{x}**not**ever touch the

*x*axis, although it comes arbitrarily close.

*n*and

*x*can be any real or complex number, or even any element of a Banach algebra or the field of

*p*-adic numbers.

To see the equivalence of these definitions, see Definitions of the exponential function.

If *x* is real, then *e ^{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:

*a*> 0 and .

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*:

*a*and

*b*and all real numbers

*x*. Expressions involving fractions and roots can often be simplified using exponential notation because:

## Exponential function and differential equations

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

The exponential function thus solves the basic differential equation

## Exponential function on the complex plane

When considered as a function defined on the complex plane, the exponential function retains the important properties

*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

*a*and

*b*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 multi-valued function ln(

*z*). We can define a more general exponentiation:

*z*and

*w*. This is then also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued 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.

## Exponential function for matrices and Banach algebras

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

*we should add the general formula involving commutators here.*)

*e*is invertible with inverse^{x}*e*^{-x}- the derivative of exp at the point
*x*is that linear map which sends*u*to*u*·*e*.^{x}

## 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.