I am looking at the following example problem dealing with complex numbers:
$$\int_{-\infty }^{\infty }\left | e^{-4t +j\frac{\pi }{4}} \right |^{2}dt$$
In this example when you take the magnitude square of a complex the integral becomes:
$$\int_{-\infty }^{\infty } e^{-8t} dt$$
I am not understanding how the imaginary part disappears when you take the magnitude square of a complex number. I understand that $j^2$ = -1 but it doesn't make sense how the imaginary part goes away. Any assistance in understanding this step is greatly appreciated.
Best Answer
For real $a,b$ we have
$$|e^{a+jb}|=|e^ae^{jb}|=e^a|e^{jb}|=e^a,$$
since $|e^{jb}|=1.$