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.
Some bits of notation. Let R\P be polar notation for R(cos(P)+i.sen(P)). Let ^ denote exponentiation.
By definition, X^Y = e^(Y.ln(X)). If the imaginary part of Y.ln(X) is zero, the result is real.
Let X=R\P; then ln(X)=ln(R)+i.P. Also, let Y=R'\P' = R'(cos(P')+i.sen(P')). The imaginary part of Y.ln(X) is then R'(ln(R).sen(P')+P.cos(P'))=0.
If we don't allow R'=0, we could take
P=-1/cos(P'), ln(R)=1/sen(P'), and we would have the sought result.
Did I miss anything? Opinions?