/sci/ - Science & Math

My book writes electric fields in the complex notation (pic related), saying that the complex conjugate ensures the quantity is real. But it also says that [math]A(r,t)[/math] can be complex, and this doesn't make sense to me because I get:
[math]Ae^{i(kz-\omega t)}+A^*e^{-i(kz-\omega t)}=(A+A^*)\cos(kz-\omega t)[/math]
Which is not always real. What am I doing wrong?