Fourier transform accumulation property

Question

It is known that Fourier Transform has an "accumulation property". If we define:

$$X(\omega) = F[x(n)]$$ and $$y(k) = \sum\limits_{n=-\infty}^k x(n)$$ i can write: $$y(k)-y(k-1)=\sum\limits_{n=-\infty}^k x(n)-\sum\limits_{n=-\infty}^{k-1} x(n)=x(k)$$ Where both $x(n)$ and $y(n)$ are discrete functions. Now, if i'm not mistaken, the accumulation property states that the Fourier Transform of $y(n)$ can be written as $$\frac{X(\omega)}{(1-e^{-i\omega})}+\pi \sum\limits_{n=-\infty}^{+\infty} X(2\pi n) \delta(\omega-2\pi n)$$

Now, I understand the second term exists to yield the result of $Y(\omega)$ when $x(n)$, but I don't understand how this writing works, because when $\omega=0$ the first term goes to infinity anyway.

My question is, shouldn't the definition be:

$$ \begin{cases} \frac{X(\omega)}{(1-e^{-i\omega})}, \omega\neq0 \\ \pi \sum\limits_{n=-\infty}^{+\infty} X(2\pi n) \delta(\omega-2\pi n), \omega=0 \end{cases} $$

because i think the other expression will always be infinite when $\omega=2 \pi n$.

Thanks in advance.

Answer 1

Consider the discrete fourier transform of the step function $u(t)$

$$\frac{1}{1-e^{i\omega}}+\pi\sum\limits_{n=-\infty}^{+\infty} \delta(\omega-2\pi n)$$

And the continuous fourier transform

$$\frac{1}{i\omega}+\pi\delta(\omega)$$

It is supposed to be like this. The formula is correct. It is important to note that the step function is not actually an absolutely integrate function, so its fourier transform is more like a principal value.