Fourier transform:

$\displaystyle X_k$	$\displaystyle = \sum_{n=0}^{N-1} x_n \, e^{-\,2\pi i\, kn/N}, \qquad k=0,1,\dots,N-1 ,$	(1)
$\displaystyle x_n$	$\displaystyle = \frac{1}{N} \sum_{k=0}^{N-1} X_k \, e^{\,2\pi i\, kn/N}, \qquad n=0,1,\dots,N-1 ,$	(2)
$\displaystyle \sum_{n=0}^{N-1} e^{\,2\pi i (k-\ell)n/N}$	$\displaystyle = N\,\delta_{k\ell}.$	(3)

$\displaystyle \hat f(k)$	$\displaystyle = \int_{-\infty}^{\infty} f(x)\, e^{-\,i k x}\,dx ,$	(1)
$\displaystyle f(x)$	$\displaystyle = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat f(k)\, e^{\,i k x}\,dk ,$	(2)
$\displaystyle \int_{-\infty}^{\infty} e^{\,i (k-k') x}\,dx$	$\displaystyle = 2\pi\,\delta(k-k') .$	(3)