Math Majors: Complex Number Exponents Help Plz.

[Bobmuhthol]

My precalc teacher brought something to my attention today that neither of us know the answer to: i^x, where x is a real integer, is very simple to evaluate; however, i^i is a decimal (~2). A real number, x, with an exponent of i will result in an answer of y + zi.

I'd love to know how to solve this, please.

[Edited on 11-28-2005 by Bobmuhthol]

[Latrinsorm]

i^i = x
(i^i)^i = x^i
i^(i*i) = x^i
i^(-1) = x^i
1/i = x^i
-i = x^i
ln(-i) = ln(x^i)
-pi*i/2 = i*ln(x)
-pi/2 = ln(x)
e^(-pi/2) = x = .207 blah blah blah

I'll try the second part later.

[Hulkein]

Just ask an Asian.

[Latrinsorm]

While not Asian, the second part turned out to be easier than I expected.

x^i = y
i * ln(x) = ln(y)
i = ln(y)/ln(x)
e^i = e^(ln(y) / ln(x))
(e^i)^(ln(x)) = y
e^(i * ln(x)) = y

Using the unit circle, cos(ln(x)) = Im(y), sin(ln(x)) = Re(y).