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Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in Kempf et al., New Dirac Delta function based methods with applications to perturbative expansions in quantum field theory): \begin{equation} \int^x f(x')\,dx' = \lim_{y \to 0} f\left(\frac{\partial}{\partial y}\right) \frac{\mathrm{e}^{xy}-1}{y} +C, \end{equation} I'm wondering whether Euler in his very imaginative calculations (to say the least) did use some techniques (in special cases) that amount to this formula.

Any hints are welcome.

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    $\begingroup$ The formula just seems to be a funny way of saying that one can integrate a power series term by term. $\endgroup$ May 18 at 20:06
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    $\begingroup$ In principle yes. But nevertheless it can help in performing integrals. At least it has been implemented in Maple from version 2019 onwards, according to uwaterloo.ca/physics-of-information-lab). $\endgroup$ May 18 at 20:19
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    $\begingroup$ Since this is a question about what Euler knew, rather than about mathematics per se, it seems like a better fit for HSMSE. $\endgroup$
    – LSpice
    May 18 at 21:41
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    $\begingroup$ Could you give an example of a specific integral which is evaluated by this formula? $\endgroup$ May 18 at 23:16
  • $\begingroup$ One of the simplest examples is $f(x) = \exp(x)$, so $f(\partial_y)g(y)=g(y+1)$ and you get by the formula above the anti–derivative $\exp(x)–1+C$. $\endgroup$ May 19 at 22:56

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I don't know about that particular integral, but Euler certainly knew about integrating by differentiating. He wrote about it in his Exposition de quelques paradoxes dans le calcul integral (1758). A recent summary of that work can be found in

A. Fabian and H.D. Nguyen, Paradoxical Euler: integrating by differentiating, The Mathematical Gazette 97 (2013), no. 538, 61-74.

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    $\begingroup$ (this is certainly far off-topic and I will delete it shortly, but I would like to say I am a big fan of your twitter account!) $\endgroup$ May 19 at 20:17

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