Let the function $ f(x)= \begin{cases} 0: &|x|>1\\ 1: &|x| \leq 1 \end{cases} $
$|x|$ is the euclidian norm of $x$.
My question is how we calculate $F(f)$? (the Fourier transformed). I tried to do this: $$\begin{eqnarray} F(f)(\xi) &= & \int_{\mathbb{R}^n} f(x) \exp \big( -i \langle x,\xi \rangle \big) dx \\ & = & \int_{[-1,1]^n} \exp \bigg( -i \sum_{k=1}^n x_k \xi_k \bigg) dx_1 dx_2 \ldots dx_n = \prod_{k=1}^n \int_{-1}^1 \exp (-i x_k \xi_k) dx_k =\Pi_{i=1}^n \int_{-1}^1 [\cos(x_i \xi_i) - i \sin(x_i \xi_i)] dx_i \end{eqnarray}$$ But how I can conclude the calculation?