[seqfan] Re: Does (e i)^(pi i) have an imaginary part?
Jack Brennen
jfb at brennen.net
Fri Aug 6 00:56:52 CEST 2010
You can "simplify" it to:
e^(pi i) * i^(pi i)
-1 * (i^i)^pi
And it is well known that i^i is a positive real number. So your number
is purely real, being the negation of a positive real number to a
real exponent.
Alonso Del Arte wrote:
> This may be a question with a very obvious response, but it seems to be a
> bit over my head. We all know e^(pi i) has no imaginary part (its real part
> famously being -1). What about (e i)^(pi i)?
>
> In Mathematica, I get a real part of
> approx. -0.007191883355826365607801366396371202955362318 and an imaginary
> part of 0. * 10^(-23) (if I ask for 20 decimal places precision) or 0. *
> 10^(-53) (if I ask for 50). Is the imaginary part so small it's beyond
> machine precision, or am I chasing a phantom?
>
> Al
>
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