I want to compute the Fourier transform for the function $f(x)=\sin{x}$. The first question is: the Fourier transform is $\pi$?
$$a_n=\frac{1}{\pi}\int^{\pi}_{-\pi}{\sin{x}\cos{nx}}=0$$ $$b_n=\frac{1}{\pi}\int^{\pi}_{-\pi}{\sin{x}\sin{nx}}=\pi,\mbox{ for }n=1 \mbox{and for the other values of }n \mbox{ is } 0$$
The second question is how can I draw the graph of frequency versus amplitude?
Thanks!
It's kind of silly:
$$\sin(x) = \frac{1}{2}a_0+\sum_{n\geq 1} a_n \cos(nx) + \sum_{n\geq 1} b_n\sin(nx).$$
Clearly, $\sin(x)=\sin(x)$, so $b_1=1$ and everything else is 0.
From the coefficient formula, just double check that $\int_{-\pi}^\pi\sin(x)^2dx=\pi$.