Engineering Tables/DTFT Transform Table

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