Originally Posted by
topsquark NOTE: I have corrected the 9:30AM version of this!
You need to split y into its real and complex parts:
I'm going to call alpha = a and beta = b and I^2 = -1. (Don't confuse these with the coefficients of the differential equation!)
y = k1*exp{a + Ib} + k2*exp{a - Ib}
y = k1*exp{a}*exp{Ib} + k2*exp{a}*exp{-Ib}
y = exp{a}*[k1*exp{Ib} + k2*exp{-Ib}]
y = exp{a}*[k1*cos(b) + I*k1*sin(b) + k2*cos(b) - I*k2*sin(b)]
Define k1 = c + Id and k2 = e + If where c, d, e, f are real.
Then
y = exp{a}*[c*cos(b) + I*d*cos(b) + I*c*sin(b) - d*sin(b) + e*cos(b) + I*f*cos(b) - I*e*sin(b) + f*sin(b)]
y = exp{a}*[c*cos(b) - d*sin(b) + e*cos(b) + f*sin(b)] + I*exp{a}*[d*cos(b) + c*sin(b) + f*cos(b) - e*sin(b)]
The complex part of y is
exp{a}*[d*cos(b) + c*sin(b) + f*cos(b) - e*sin(b)]
= exp{a}*[(d + f)*cos(b) + (c - e)*sin(b)]
This must be 0. To be true for all b we must have that
d + f = 0
c - e = 0
Thus f = -d
Thus e = c.
So k2 = e + If = c - Id, which is the complex conjugate of k1 = c + Id.
-Dan