The term Poisson integral formula may refer to any of the related formulas for harmonic (or holomorphic) functions on a disk (or in a ball, half space, etc) in terms of their boundary values. This is the formula that contains the famous Poisson kernel. When and where did Poisson publish his integral formula? I have a paper which says it was published in 1820 (without reference) and hours of searching has revealed nothing more.
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I think that this is what you are asking for:
Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, et sur l’usage de cette transformation dans la résolution de différents problèmes, Journal de l'École polytechnique, $18^e$ cahier, 11 (1820), p. 417–489.
Here you can find the scan of the original paper (in French, of course).
EDIT
The so-called "noyau de Poisson" (Poisson kernel, clearly Poisson does not call it this way) appears at page 422.
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1$\begingroup$ Great! It is interesting that nothing is mentioned of the Laplace equation, Dirichlet problem etc but the Poisson kernel appears as a regularization of the Fourier series representing the delta measure. $\endgroup$– timurDec 23, 2021 at 21:25
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$\begingroup$ ... Although Fourier will publish his "Théorie Analytique de la Chaleur" (where the Fourier series are introduced) only in 1828. $\endgroup$ Jan 4, 2022 at 0:32
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1$\begingroup$ @JeanMarieBecker Fourier series appear for the first time in "Mémoire sur la propagation de la Chaleur dans les corps solides", pp. 112-116, Nouveau Bulletin des Sciences par la Société Philomathique, published in March 1808 but presented at a session of the French Academy on the 21st of December 1807 at the presence of Lagrange, Laplace, Monge and Lacroix $\endgroup$– user6530Jan 4, 2022 at 15:03
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1$\begingroup$ @user6530 Thank you very much for this important precision. $\endgroup$ Jan 4, 2022 at 15:07