You can go from the second equation to the third by assuming A= -2aj and B= -2aj(j)= 2a. But when I substitute that into the last equation, I get zero, not 2a. Either I'm doing that wrong, or one of the last three equations is wrong.
This is the equation it comes from originally. I was thinking that if -j rotated the complex clockwise by 0.5pi vectorialy but even then the components don't add up since j is a necessary coefficient in de moviere theorem!
Another issue is getting rid of j when the terms inside the bracket are multiplied by j which eliminates j on sine term but not on the cosine one.
The amplitude is bothersome indeed.
so e^(-ikx) - e^(ikx) = cos(-kx) + i*sin(-kx) - (cos(kx) + i*sin(kx)) = (cos(-kx) - cos(kx)) + i*(sin(-kx) - sin(kx))
cos(-kx) = cos(kx)
sin(-kx) = -sin(kx)
so we get = -2i*sin(kx) which is what you have.
another day in paradise, or is paradise one day closer ?
I have a gut feeling the author is using complex vector geometry in the argand plane or in polar coordinates. I will think into it otherwise I am moving on. It will be bothersome for a bit.