Real part of a product?

Question

We have: $$ \Re\{Ae^{i\phi}e^{i\omega}\}=\Re\{Ae^{i(\phi+\omega)}\}=A\cos(\phi+\omega) $$ But why does the following not give the same the answer: \begin{align} \Re\{Ae^{i\phi}e^{i\omega}\} &=A\Re\{e^{i\phi}e^{i\omega}\}\\ &=A\Re\{e^{i\phi}\}\Re\{e^{i\omega}\}\\ &=A\cos(\phi)\cos(\omega) \quad ? \end{align}

Answer 1

You made the assumption in the penultimate line that the real part of a product is a product of the real parts. But that's simply not true. Remember the definition for complex multiplication in rectangular coordinates?

Say let us have $w=a+ib$ and $z=p+iq$, then $$\Re(wz)=ap-bq\neq{ap}=\Re(w)\Re(z)$$ as you assumed.

Answer 2

As @CaveJohnson wrote in a comment, the real part of a complex number is not a multiplicative function. An easy counter-example would be

$$\Re(i)\Re(i)=0\neq-1=\Re(i^2)$$

Hence, only your first calculation is correct.

EDIT: You can calculate directly and see that in fact, if $z_{1}=a+bi$ and $z_{2}=c+di$, then

$$\Re\left(z_{1}z_{2}\right)=\Re\Big((ac-bd)+(ad+bc)i\Big)=ac-bd=\Re(z_{1})\Re(z_{2})-\color{red}{\Im(z_{1})\Im(z_{2})}$$