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From: zephyrtronium on 2 Jun 2010 22:41
I was playing around on my calculator and I happened to find that
apparently (-1) ^ (x / pi) = e ^ (x i). Then I got even more bored and
I decided to see what (-e) ^ (x / pi) would look like, and then I got
still more bored and decided to find a value a such that a ^ x e ^ (x
i) = (-e) ^ (x / pi). That's when I ran into troubles. I've tried all
my favorite numbers but none of them are exactly correct; e / 2 is the
closest I've found. Now this problem is bugging me, so can anyone help
me find that value?

Sorry if this is in the wrong place or anything.
From: Tim Little on 2 Jun 2010 23:30
On 2010-06-03, zephyrtronium <zephyrtronium(a)gmail.com> wrote:
> I was playing around on my calculator and I happened to find that
> apparently (-1) ^ (x / pi) = e ^ (x i).

If you choose an appropriate branch of the complex logarithm, it
follows from Euler's identity e^(i pi) + 1 = 0.


> decided to find a value a such that a ^ x e ^ (x i) = (-e) ^ (x / pi).

Given a value for x, it is not very difficult to find a corresponding
value for 'a' that works. Do you expect there to be a single value of
a that works for all x?


- Tim
From: zephyrtronium on 3 Jun 2010 00:04
On Jun 2, 11:30 pm, Tim Little <t...(a)little-possums.net> wrote:
> On 2010-06-03, zephyrtronium <zephyrtron...(a)gmail.com> wrote:
>
> > I was playing around on my calculator and I happened to find that
> > apparently (-1) ^ (x / pi) = e ^ (x i).
>
> If you choose an appropriate branch of the complex logarithm, it
> follows from Euler's identity  e^(i pi) + 1 = 0.

That makes sense... We never even mentioned Euler's law in precalc, so
I have difficulties grasping some of its implications.

> > decided to find a value a such that a ^ x e ^ (x i) = (-e) ^ (x / pi)..
>
> Given a value for x, it is not very difficult to find a corresponding
> value for 'a' that works.  Do you expect there to be a single value of
> a that works for all x?

I was hoping for a single value. It would put my mind to rest.
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