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Re: What's the geometry meaning of i^i ?
Posted:
Jul 26, 2012 8:18 AM
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Am 26.07.2012 07:09 schrieb Hongyi Zhao:
> On Thu, 26 Jul 2012 05:21:56 +0200, Gottfried Helms wrote:
>
>> ... using rectangular coordinates and interpret them as polar ones (in
>> principle)?
>
> What do you mean, could you please give some more detail explanations?
>
> Regards
>
Hmm. Assume z0 = a + b*i where i is the imaginary unit.
Then consider
z1 = e^z0 = exp(z0) = e^a * e^(bi)
and e^a defines the length and e^(bi) the arc of the vectorial
representation of z1. Assume then new letters for the rectangular
coordinaters of z1:
z1 = A + B*i
Then consider the iteration
z2 = e^z1 = e^A * e^(Bi)
So the exponentiation can be understood as re-intrpretation
of the rectangular coordinates as coordinates in a polar representation.
Using i instead of e as basis for the exponentiation adds
a bit of complication, since i = e^(Pi/2*i), but that
complication is not so weird...
Gottfried
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