Exponential function: Difference between revisions

Revision as of 12:06, 1 September 2001

Revision as of 18:19, 27 November 2001

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exp(''x'') = ∑ ---

''n''=0 ''n''!

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''n''→∞

(see [[limit]] and [[Infinite Series|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]].

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

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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]]s spring from the exponential function in one way or another.

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:''a''⁰ = 1

:''a''^{''x'' + ''y''1} = ''a''^''x'' ''a''^''y''

:''a''^{(''xyx'' + ''y'')} = (''a''^''x'') ''a''^''y''

:''a''^(''xxy'') = (''ba''^''x'' = (''ab'')^''xy''

:1 d/d''x'' ''a''^''bxx'' = ln(1/''a'') ^''bx'' = ''a''^-''bxx''.

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

These are valid for all positive real numbers ''a'' and ''b'' and all real numbers ''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

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

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

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

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

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

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.

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

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

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.

exp'(''z'') = exp(''z'')

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

and

:exp(''z'' + ''w'') = exp(''z'') exp(''w'')

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

:exp(''z'') &ne; 0

:exp'(''z'') = exp(''z'')

for all ''z'' and ''w''. The exponential function on the complex plane

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:

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:''z''^''w'' = exp(ln(''z'') ''w'')

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:''f''(''t'') = exp(''t'' ''A'')

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:''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 hasshares the samethese properties, which explains the terminology.