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Exponential function as a special case of Lie algebra exponential maps |
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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.
=== Exponential function on the complex plane ===
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▼
▲When considered as a function defined on the [[complex number|complex plane
:exp(''z'' + ''w'') = exp(''z'') exp(''w'')
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This is then also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.
=== Exponential function for matrices and Banach algebras ===
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:▼
The definition of the exponential function exp given above can be used verbatim for every [[Banach algebra]], and in particular for square [[matrix|matrices]]. In this case we have
:exp(''x'' + ''y'') = exp(''x'') exp(''y'') if ''xy'' = ''yx'' (''we should add the general formula involving commutators here.'')
:exp(0) = 1
: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(''x'')·''u''.
▲In the context of
:''f''(''t'') = exp(''t'' ''A'')
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:''f''(''s'' + ''t'') = ''f''(''s'') ''f''(''t'')
:''f''(0) = 1
:
=== 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.
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