[seqfan] Re: Does (e i)^(pi i) have an imaginary part?

[Charles Greathouse]

(e i)^(pi i) = e^(pi i) * i^(pi i) = -1 * exp(pi i log i) = -exp(pi i
* pi/2 i) = -exp(-pi^2/2) = -0.00719188335582636560780136639...

So yes, no imaginary part.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Thu, Aug 5, 2010 at 6:34 PM, Alonso Del Arte
<alonso.delarte at gmail.com> 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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