The following formulas give natural differintegral (that is one with naturally fixed integration constant):

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

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

When $s=-1$ the expressions produce the antiderivative of the function $f(x)$. I wonder, whether there are any practical applications where the integration constant fixed this way is useful? For instance, unsurprisingly it gives $\sin(x+\frac{\pi s}2)$ as differintegral of sine, but non-trivially gives $\ln(x)+\gamma-i\pi=\ln(-xe^{\gamma})$ as integral of $1/x$

I also wonder whether $\ln(xe^{\gamma})$ has any intuitive meaning (especially given the [analogy][1] between $e^{-\gamma}$ and $\frac{\pi}4$).

  [1]: https://mathoverflow.net/questions/341470/is-there-any-deep-philosophy-or-intuition-behind-the-similarity-between-pi-4