I am reading about Spectral Analysis from this set of notes, and I have some doubts regarding the algebraic manipulation. Since I am kinda new in this topic I don't really know if I'm missing something or there's a typo.
They start defining $X_t=\sum_jA_j\cos(2\pi\lambda_jt)+B_j\sin(2\pi\lambda_jt)$ (where X is a real-values random variable), and then they re-express it using the complex representation: $\sum_{\pm j}Ce^{2\pi i\lambda_jt}$ where $C_{-j}=C^*_j$ (since $X$ is real valued) and $\lambda_{-j}=-\lambda_j$
Then they give the following definition $C_j=A_j+iB_j$
Taking one term of the summation we can re express it as:
$$(A_j+iB_j)[\cos(2\pi\lambda_jt)+i\sin(2\pi\lambda_jt)]+(A_j-iB_j)[\cos(-2\pi\lambda_jt)+i\sin(-2\pi\lambda_jt)]=$$ $$\cos(2\pi\lambda_jt)[A_j+iB_j+A_j-iB_j]+i\sin(2\pi\lambda_jt)[A_j+iB_j-A_j+iB_j]=$$ $$2A_jcos(2\pi\lambda_jt)-2B_j\sin(2\pi\lambda_jt)$$
But it's easy to see that actually the coefficients are different (multiplied by $2$) and we have a minus instead of a plus.
My question is, did the authors considered $C_j=A_j+iB_j$ as a way to say that $C_j$ is a complex number, or were they making reference to the coefficients of the first expression? The notation is somehow ambiguous.