These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).

The first one is based on Newton series, interpolated over consecutive derivatives:

$$f^{(-1)}(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$

The second one is based on Furier transform:

$$f^{(-1)}(x)=\frac{i}{2\pi}\int_{-\infty}^{+\infty} \frac{e^{- i \omega x}}{\omega} \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, d\omega$$

The question is for a proof that the both definitions coincide exactly (i.e. their values at all points and not only up to a constant) when the both converge.

Without losing the generality it is possible to consider only one point, say, $x=0$, since equality at this point guarantees equality elsewhere.

Note. $f(x)$ is required to be alalytic.

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