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"See also" is NOT just a part of the section titled "Exponential map on Lie algebras" |
Making the sections first-level sections |
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: <math>\sqrt[c]{a}^b = a^{b \over c}</math>
The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own [[Derivative|derivatives]]:
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and it is for this reason commonly encountered in differential equations. In particular the solution of linear ordinary [[differential equation]]s 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]].
When considered as a function defined on the [[complex number|complex plane]], the exponential function retains the important properties
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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]].
The definition of the exponential function exp given above can be used verbatim for every [[Banach algebra]], and in particular for square [[matrix_(mathematics)|matrices]]. In this case we have
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: <math>f'(t) = A f(t)</math>
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(<i>n</i>, '''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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