Engineering Tables/DTFT Transform Table

Time domain $x[n]\,$ where $n\in \mathbb {Z}$	Frequency domain $X(e^{j\omega })$ where $\omega \in \mathbb {R}$	Remarks
${\frac {1}{2\pi }}\int _{-\pi }^{\pi }{X\left(e^{j\omega }\right)}e^{j\omega n}d\omega$	$\sum _{n=-\infty }^{\infty }{x[n]e^{-j\omega n}}$	Definition
$x[n]={\begin{cases}1,&|n|\leq M\\0,&{\text{otherwise}}\end{cases}}$	${\frac {\sin \left(\omega \left({\frac {2M+1}{2}}\right)\right)}{\sin \left({\frac {\omega }{2}}\right)}}$
$\alpha ^{n}u\left[n\right]$	${\frac {1}{1-\alpha e^{-j\omega }}}$
$\delta [n]$	$1\!$	Here $\delta [n]$ represents the delta function which is 1 if $n=0$ and zero otherwise.
$u[n]={\begin{cases}0&{\text{for }}n<0\\1&{\text{for }}n\geq 0\end{cases}}$	${\frac {1}{1-e^{-j\omega }}}+\pi \sum _{p=-\infty }^{\infty }{\delta \left(\omega -2\pi p\right)}$
${\frac {1}{\pi n}}\sin \left(Wn\right),\;\;\;\;0<W\leq \pi$	$X(e^{j\omega })={\begin{cases}1,&|\omega |\leq W\\0,&W<|\omega |\leq \pi \end{cases}}$	$X(e^{j\omega })$ is 2π periodic
$(n+1)\alpha ^{n}u\left[n\right]$	${\frac {1}{(1-\alpha e^{-j\omega })^{2}}}$