These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant ([see this question for further details][1]).

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:

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

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.


  [1]: http://math.stackexchange.com/questions/902107/is-there-an-easier-way-to-find-the-natural-integration-constant