Fourier Transform of char. function of $d$-dimensional unit cube

Question

I want to find the Fourier transform of the unit cube. So far, I have $$f(\xi) = \frac{1}{(2\pi)^\frac d 2}\int_{\mathbb{R}^d}\chi_{[-1,1]^d}e^{-i\langle x,\xi\rangle}dx = \frac{1}{(2\pi)^\frac d 2}\int_{[-1,1]^d}e^{-i\langle x,\xi\rangle}dx$$ Now I don't know how to continue with that dot product in the exponent, any help would be appreciated.

Answer 1

Note that since $\langle x,\xi\rangle=\sum_{n=1}^d x_n\xi_n$ we have

$$\int_{[-1,1]^d}e^{-i\langle x,\xi\rangle}\,dx=\prod_{n=1}^d \int_{-1}^1 e^{-ix_n\xi_n}\,dx_n$$

And now you can finish.