Interpreting Fourier Series

Question

I came accross an interpretation of complex fourier series as vector sum of vectors described by complex numbers $ e^{inx}$, where n ranges over integers.

Although I understand this form but I am not able to account for the following points:

1) While transitioning from cosine or sine wave to complex, it is just the real part that makes sense, right? (This is in context to the interpretation and not the formal proof)

2) If point 1 is true, why don't we take just the real part of the series while writing it? Are we sure that the series always gives out real value?

Answer 1

Yes, the final result is real.

In fact the big difference with Fourier series expressed with real coefficients $a_n$ and $b_n$ is that indices $n$ of the $c_n$ coefficients, instead of varying between $0$ (or $1$) and $+\infty$ vary between $-\infty$ and $+\infty$. Therefore, opposite index coefficients $c_{-n}$ and $c_n$ are conjugate one to the other, therefore "cooperate" to yield finally real numbers.

Indeed, from the formulas :

$$c_0=constant, \ \ \ c_n=\tfrac12(a_n-ib_n), \ \ \ \ \ c_{-n}=\tfrac12(a_n+ib_n) \ \ \text{for} \ n>0$$

one deduces :

$$c_ne^{inx}+c_{-n}e^{-inx}=\tfrac12(a_n-ib_n)e^{inx}+\tfrac12(a_n+ib_n)e^{-inx}=\tfrac12 \left(a_n\underbrace{(e^{inx}+e^{-inx})}_{2 \cos x}-ib_n\underbrace{(e^{inx}-e^{-inx})}_{2i \sin x} \right)$$

and we find back in this way the usual formulas.

See the nice animations there explaining the interest of considering complex Fourier coefficients