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Exponential function

The exponential function is one of the most important functions in mathematics. It is written as exp(x) or <math>e^x</math> (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 ex does not ever touch the x axis, although it comes very close.

<math>\exp(x) = \sum_{n = 0}^{\infty} {x^n \over n!}</math>
<math>\exp(x) = \lim_{n \to \infty} \left( 1 + {x \over n} \right)^n</math>

Here <math>n!</math> stands for the factorial of <math>n</math> and <math>x</math> can be any real or complex number, or even any element of a Banach algebra or the field of p-adic 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:

<math>a^x = \exp(\ln(a) x)</math>
for all <math>a > 0</math> and all real <math>x</math>.

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:

<math>a^0 = 1</math>
<math>a^1 = a</math>
<math>a^{x + y} = a^x a^y</math>
<math>a^{x y} = \left( a^x \right)^y</math>
<math>{1 \over a^x} = \left({1 \over a}\right)^x = a^{-x}</math>

<math>a^x b^x = (a b)^x</math>

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

<math>{1 \over a} = a^{-1}</math>
<math>\sqrt{a} = a^{1/2}</math>
<math>\sqrt[n]{a} = a^{1/n}</math>

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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:

<math>{d \over dx} a^{bx} = \ln(a) b a^{bx}.</math>

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

<math>{dy \over dx} = y</math>
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.

Exponential function on the complex plane

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

<math>\exp(z + w) = \exp(z) \exp(w)</math>
<math>\exp(0) = 1</math>
<math>\exp(z) \ne 0</math>
<math>\exp'(z) = \exp(z)</math>
for all z and w. The exponential function on the complex plane is a holomorphic function which is periodic with imaginary period <math>2 \pi i</math> which can be written as

<math>\exp(a + bi) = \exp(a) \cdot (\cos(b) + i * \sin(b))</math>

where <math>a</math> and <math>b</math> 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:

<math>z^w = \exp(\ln(z) w)</math>
for all complex numbers 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

<math>\exp(x + y) = \exp(x) \exp(y)</math>
if <math>xy = yx</math> (we should add the general formula involving commutators here.)
<math>\exp(0) = 1</math>
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(xu.

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:

<math>f(t) = \exp(t A)</math>
where <math>A</math> is a fixed element of the algebra and <math>t</math> is any real number. This function has the important properties
<math>f(s + t) = f(s) f(t)</math>
<math>f(0) = 1</math>
<math>f'(t) = A f(t)</math>

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.

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