Is it possible for two complex numbers to have a real exponentiation?
In other words, if x and y are complex (each have the form a+bi), show that x^y can have a real value, or prove that it is impossible.
Note: i is the imaginary value defined as the number that yields -1 when squared. a and b are any real numbers, but b is not 0.
(In reply to
re(3): solution by Richard)
Ok, we all know e^(iπ)=-1, right?
So, e^(2iπ)=1
e^(2iπ*i)=1^i
e^(-2π)=(1^0)^i
e^(-2π)=1^0=1
So, e^(-2π)=1
Yeah, that last equation is obviously incorrect. This little paradox comes from the fact that that law of exponents doesn't work with complex numbers with magnitude greater than e^π... or something like that. There's some calculus reason behind it, but I don't really understand it.
Edited on November 27, 2003, 1:54 pm
Edited on November 27, 2003, 1:56 pm
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Posted by Tristan
on 2003-11-27 13:52:19 |
re(4): solution
