I have the following expression:
$$\frac{(\cos x + i \sin x) (\cos nx + i \sin nx)(\cos x + i \sin x)}{1-(\cos x + i \sin x)}$$
I need to separate the imaginary and real parts however I am not understanding how should I deal with the denominator.
Question:
What should I do to separate the expression into real and imaginary parts? Am I missing some complex number property?
Thanks in advance!
Best Answer
If you have ${a\over b}$ where $a,b$ are complex, then ${a \over b} = {a \overline{b} \over |b|^2}$.
Then ${a \over b} = {1 \over |b|^2} (\operatorname{re} (a \overline{b}) + i\operatorname{im} (a \overline{b}) )$.
In your case, the expression is ${e^{ix} e^{inx} e^{ix} \over 1-e^{-ix}} = {e^{i(2+n)x} (1-e^{-ix})\over (1-e^{-ix}) (1-e^{-ix})} = {e^{i(2+n)x} (1-e^{-ix})\over 2(1-\cos x)}$.
Hence we have ${1 \over 2(1-\cos x)} (\cos((2+n)x)) - \cos((1+n)x) + i(\sin((2+n)x)) - \sin((1+n)x)))$.