get the n knowing Fourier coefficients

Question

Given a generic Fourier series and knowing that $a = \dfrac{-2}{3}$ and $b = \dfrac{-2}{3}$ and that $n\ge 1$, how to obtain the exponential representation, i.e. complex coefficients and the power of $e$?

Answer 1

You can use the following result: $$c_n=\frac{a_n}{2}-i\frac{b_n}{2}\,\,\text{for}\,n\ne0$$ to obtain you answer. To see this: write $$a_n \cos (n\omega t) + b_n \sin (n\omega t) = c_{- n} e^{-in\omega t} + c_n e^{in\omega t}\quad\text{for}\,n\ne0$$ because of the fact that real and complex representations are equivalent. Since $c_n=c_{-n}^*$, after simplifying you'll have the result.