Exponential function: Difference between revisions

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

∞ ''x''^''n''

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 number|real]] or [[complex number|complex]] number, or even any element of a [[Banach algebra]] or the field of [[p-adic numbers|<i>p</i>-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:

:''a''^''x'' = exp(ln(''a'') ''x'')

for all ''a'' > 0 and all real ''x''.

The exponential function also gives rise to the [[trigonometric function|trigonometric functions]] (as can be seen from [[Eulers formula in complex analysis|Euler's formula]]) and to the [[hyperbolic function|hyperbolic functions]]. Thus we see that all elementary functions except for the [[polynomial|polynomials]] spring from the exponential function in one way or another.

The exponential function also gives rise to the [[trigonometric functions]] (as can be seen from [[Eulers formula in complex analysis|Euler's formula]]) and to the [[hyperbolic functions]]. Thus we see that all elementary functions except for the [[polynomial|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''⁰ = 1

:''a''¹ = ''a''

:''a''^{''x'' + ''y''} = ''a''^''x'' ''a''^''y''

:''a''^(''xy'') = (''a''^''x'')^''y''

:1 / ''a''^''x'' = (1/''a'')^''x'' = ''a''^-''x''

:''a''^''x'' ''b''^''x'' = (''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

:&radic; ''a'' = ''a''^1/2

:^''n''&radic; ''a'' = ''a''^1/''n''

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

:d/d''x'' ''a''^''bx'' = ln(''a'') ''b'' ''a''^''bx''.

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'') &ne; 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 2&pi;''i'', 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:

:''z''^''w'' = 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 space|Banach]] or [[Hilbert space|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/d''t'' ''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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[[talk:Exponential_function|/Talk]]

/Talk