I have the Fourier Transform of the discrete sequence $u_j, j = 0, \pm 1, \pm2, ...$: $$ \hat{u}(\xi h) = \frac{h}{2\pi}\sum_{j=-\infty}^{\infty}u_je^{-ij\xi h} $$
and wish to verify that the inverse transform is $$ u_j = \int_{-\pi/h}^{\pi/h} \hat{u}(\xi h)e^{ij\xi h}d\xi $$
To do that, I try to insert the first form into the second: $$ \begin{align} u_j &= \int_{-\pi/h}^{\pi/h} \frac{h}{2\pi}\left(\sum_{k=-\infty}^{\infty}u_ke^{-ik\xi h}\right)e^{ij\xi h}d\xi\\ &= \frac{h}{2\pi}\int_{-\pi/h}^{\pi/h} \left(\sum_{k=-\infty}^{\infty}u_ke^{i(j-k)\xi h}\right)d\xi\\ \end{align} $$ I've tried expanding the integrand a few times, though haven't landed on anything useful which gives the integration yielding $hu_j/2\pi$, as it should. What must be done?