I have two complex numbers in the form $$U_{1} = Ae^{j \phi}$$ and $$ U_{2} = Be^{j \omega}$$ where $j$ is the imaginary unit.
What is the expansion of $$|U_{1} + U_{2}|^{2}\;?$$
My initial attempt was to use the FOIL method, and that yielded $$(U_1 + U_2^*)(U_1^* + U_2) = U_{1} U_1^* + U_1 U_2 + U_1^* U_2^* + U_2^* U_{2}$$ $$ = A^2 + B^2 + ABe^{j(\phi + \omega)} + ABe^{-j(\phi + \omega)}.$$
This doesn't line up with the answer I've found online, which gives the result as $$ |U_{1} + U_{2}|^{2} = U_{1} U_1^* + U_1^* U_2 + U_1 U_2^* + U_2^* U_{2}$$ $$ = A^2 + B^2 + ABe^{j(\phi - \omega)} + ABe^{j(\omega - \phi)}.$$
Where am I going wrong?
$|z|^2=zz^\ast$.
Take $z=U_1+U_2$ to get $|U_1+U_2|^2=(U_1+U_2)(U_1+U_2)^\ast=(U_1+U_2)(U_1^\ast+U_2^\ast)$.
Can you take it from here?