How to calculate complex Fourier coefficient?

Question

Consider a step function $u:[0,1] \to R $ where R is the real numbers $$ u(x) = \begin{cases} 0, & \text{if $u<1$} \\ 1, & \text{if $u\ge 1$ is odd} \end{cases} $$ Determine the complex Fourier coefficients $u_j$ s.t. $$u(x)= \sum_{-\infty}^\infty u_je^{2\pi jix} $$ Note that $$ u_j = \int_0^1 u(x)e^{-2\pi jix}dx$$ Thank you

Answer 1

$$u_j= \frac{e^{-\pi ji}-e^{2\pi ji}}{2\pi ji} = -\frac{i}{2\pi j}(\cos(-\pi j) + i\sin(-\pi j) - \cos(2\pi j) - i\sin(2\pi j)) = \cdots$$

Answer 2

Are you bound to demonstrate it with Fourier series or you can use Fourier transformation? If the second option is possible, take a look just to time domain shifting property plus step Fourier transformation

Edit: do you know how to define a sin as complex number? sin(z) = ( exp(iz) - exp(-iz))/(2*i) Just try to define your uj as it ...