What is x^i? How can you rewrite it?

[nhmllr]

Very simple question... What is x^i? How can you rewrite it?
All I could figure out is that (x^i)^i = 1/x, but that doesn't help much
Wolfram Alpha gave me this graph (real part in blue, imaginary in orange)
http://www4c.wolframalpha.com/Calculate/MSP/MSP17119i95eid65h0gce900001e7b96h101dd87d6?MSPStoreType=image/gif&s=62&w=320&h=119&cdf=RangeControl
Which is a very strange graph.

What happens?

 

[Simon Bridge]

It is probably clearer if you look at it in the complex plane.
Apart from that, what happens is exactly what the graph says happens.

consider:
$e^{i\theta}$ is just the unit vector rotated anti-clockwise in the complex plane by $\theta$ radiens.

$a^b = e^{b\ln{a}}$ so $x^i = e^{i\ln{x}}$ so $x^i$ is the unit vector rotated by ln(x) radiens in the complex plane.

 

[nhmllr]

It is probably clearer if you look at it in the complex plane.
Apart from that, what happens is exactly what the graph says happens.

Okay, so wolfram alpha says that 3^i is about 0.455 + 0.890i
How did it figure that out?

 

[Simon Bridge]

Ah - you posted while I edited: that's a bad habit of mine.
It's a rotation in the complex plane.
The real part is the cos(ln(x)) and the imaginary part is sin(ln(x))

 

[nhmllr]

That made so much more sense than I expected it to.
It also explains this graph of y=Re(x^i)^2+Im(x^i)^2
http://www4b.wolframalpha.com/Calculate/MSP/MSP237219i95h4480ahf33i00001h6c277de8811fe7?MSPStoreType=image/gif&s=34&w=307&h=136&cdf=RangeControl
Friggin' beautiful.

 

[Simon Bridge]

Yep - when you get used to rotating phasors lots of things get simple.
I dredged up a link for you. It covers the whole imaginary exponent thing (like what happens when you raise a complex number to the power of another complex number) then links to a bunch of applications.

It's also used in analyzing linear networks (electronics) and anything with waves.