Exponential function

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The exponential function is one of the most important functions in mathematics. It is written as exp(x) or ex (where e is the base of the natural logarithm) and can be defined in two equivalent ways:

xn
   exp(x)  =  ∑  ---
             n=0  n!


   exp(x)  =  lim  (1 + x/n)n
              n→∞

(see limit and infinite series). Here n! stands for the factorial of 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.

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:

ax = exp(ln(a) x)

for all a > 0 and all real x.

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:

a0 = 1
a1 = a
ax + y = ax ay
a(xy) = (ax)y
1 / ax = (1/a)x = a-x
ax bx = (ab)x

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

1 / a = a-1
a = a1/2
na = a1/n

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

d/dx abx = ln(a) b abx.

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.

When considered as a function defined on the complex plane (or even on a commutative Banach algebra or the p-adic numbers), the exponential function retains the important properties

exp(z + w) = exp(z) exp(w)
exp(0) = 1
exp(z) ≠ 0
exp'(z) = exp(z)

for all z and w. The exponential function on the complex plane is a holomorphic function which is periodic with imaginary period 2πi, and this is the reason that extending the natural logarithm to complex arguments naturally yields a multi-valued function ln(z). We can define a more general exponentiation:

zw = exp(ln(z) w)

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.

In the context of general non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is typically considered to be a function of a real argument:

f(t) = exp(t A)

where A is a fixed element of the algebra and t is any real number. This function has the important properties

f(s + t) = f(s) f(t)
f(0) = 1
d/dt f(t) = A f(t)

The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares these properties, which explains the terminology.


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