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.
restatement of problem:
If we let x = a ^ b
where a and b are complex numbers, then prove that x can be real (or show an example where X is real)
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if a = b = i (the imaginary square root of -1), then
x = (√-1) ^ (√-1)
x = e ^ [ ln( (√-1) ^ (√-1) ) ]
x = e ^ [ (√-1) * ln (√-1) ]
x = e ^ [ (√-1) * (π/2) * (√-1) ],
since ln (√-1) = (π/2) * (√-1)
x = e ^ [ -(π/2) ]
which is real (though irrational).
example of said case
