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and Alexander Macfarlane had all employed in their writings an equivalent mapping of the Cartesian plane by operator , where is the hyperbolic angle and .

In GL(n,C)

An important example in the theory of Lie groups arises when is taken to be , the group of invertible matrices with complex entries. In that case, a basic result is the following:
:Theorem: Suppose is a one-parameter group. Then there exists a unique matrix such that
::
:for all .It follows

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