Exponential function: Difference between revisions

Revision as of 10:41, 28 March 2002

Revision as of 19:26, 6 July 2002

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for all ''z'' and ''w''. The exponential function on the complex plane is a [[holomorphic function]] which

is periodic with imaginary period 2π''i'', and this is the reason that extending the natural logarithm to complex arguments naturally yields a multi-valued function ln(''z''). Wewhich can define a more general exponentiation:

be written as

:exp(''a'' + ''bi'') = exp(''a'') * (cos(''b'') + i * sin(''b''))

where ''a'' and ''b'' are real values. This formula

connects the exponential function with the [[trigeometric function]]s, 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'')