I am trying to solve the following question:
I have tried to make an attempt at this question but I am not sure where to begin.
My current understanding is to apply $f(x) = \frac{1}{2 \pi} \int_{-\pi}^{\pi} \sin(x)e^{-inx} dx$ to find the $c_n$ coefficient for the general formula of a complex Fourier series $\sum_{n = -\infty}^{\infty} c_ne^{inx}$.
Is this the correct approach to this question?
Any help, clarification or suggestions will be greatly appreciated so I begin to solve this question. Thank you!
EDIT: attempted to evaluate $\int sin(x)e^{-ix} dx$, as a start (dont worry i did not forget about $e^{-inx}$ to obtain $\frac{-ix}{2} – \frac{1}{4} e^{-2ix} + c$ but still not sure if this is correct.
Best Answer
$$\hat{f}(n) = \frac{1}{2\pi}\int_{0}^{2\pi} f(x) e^{-inx} dx$$
$$= \frac{1}{2\pi} \int_{0}^{\pi} \sin(x) e^{-inx} dx$$
Now writing $\sin x = \frac{e^{ix} -e^{-ix}}{2i}$, we can obtain $\hat{f}(n)$.
Once we have that we can write the Fourier series as:
$$\sum_{n=-\infty}^{\infty} \hat{f}(n) e^{inx}$$