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In Riemann's "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe", Riemann mentions taking a factor as "constant" in "partial integration", which seems to me to be referring to the factor $u$ in the standard integration by parts formula $\int u\,dv = uv-\int v\, du$:

...durch zweimalige partielle Integration, indem man zuerst $\lambda(x)$, dann $\lambda'(x)$ als constant betrachtet... (pp. 23-24 here, a TeX transcription; or pp. 30-31 here, reprinted from Riemann, Abh. Gesell. der Wiss. zu Göttingen  13,  87-132  (1867)

Does anyone know where the terminology for "partial integration" is discussed? Why is a function, which is clearly not a constant function, called "constant"? Is there a reference where this is discussed?

I have a theory that it comes from a partial-derivative form of the formula for integration by parts, namely, $$\int {\partial\over\partial v}(uv)\;dv = uv-\int {\partial\over\partial u}(uv)\; du \,,$$ since in ${\partial\over\partial v}(uv)$, the factor $u$ is treated as constant. But I made that formula up and have not found it anywhere. Indeed, googling through the 19th century, the standard approach to integration by parts of today seems dominant in Cauchy and in English calculus textbooks.

Update

Replacing $uv$ by $f(u,v)$ in the above equation seemed an obvious, if unfruitful, move. Zermelo made a similar remark to Brendel, who applied it to various integrals in "Ueber partielle Integration", Math. Ann. 55, 1901 (248-256), half a century after Riemann wrote his paper. While Brendel starts with the standard integration by parts formula above, he switches the roles of $u$ and $v$ without comment and treats $v$ "als constant"; he then replaces $u$ by $x$. Be that as it may, it seems to suggest the view was what I surmised above, and presumably that was Riemann's view, too. It leaves open the question of when and where such a view of integration by parts or partial integration, in which one factor is treated as constant, arose and was common.

(By the way, Brendel remarks that Hilbert suggested that the standard formula should be called "integration of a product." (249))

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  • $\begingroup$ Is there a reason for the downvote? $\endgroup$
    – Michael E2
    Commented Mar 24, 2022 at 15:00
  • $\begingroup$ It's unclear to me, if this question is about the difference in terminology between "integration by parts" and "partial integration" (the title question) or about the origin of the terminology "partial integration". But what I can tell you is that in German, only the terminology "partielle integration" is used. The translation of "integration by parts" would be something like "integration in Teilen", which isn't used. $\endgroup$ Commented Mar 25, 2022 at 13:38
  • $\begingroup$ About the title question: I don't see a big difference in the purely linguistic meaning between "partial integration" and "integration by parts". $\endgroup$ Commented Mar 25, 2022 at 13:40
  • $\begingroup$ Thanks, @MichaelBächtold, that's a helpful confirmation. The terminology I'm asking about is the term "constant," which I've never seen used in English in integration by parts, which is always presented as the inverse of the product rule. Is it still used in German? If so, is the name justified in terms of partial differentiation or some other way? [BTW, in de.wikipedia.org/wiki/Partielle_Integration lists "teilweise Integration, Integration durch Teile" plus a Latin one as alternatives, which I understood to be less common or even much less.] $\endgroup$
    – Michael E2
    Commented Mar 25, 2022 at 14:13
  • $\begingroup$ @MichaelBächtold Thanks, too, for pointing out how the title was misleading. (Though, I guess, originally, I was wondering if there were a "partial integration" such as what Brendlel described, one that is more general in which the current form applied to products $uv$ is simply a special case.) $\endgroup$
    – Michael E2
    Commented Mar 25, 2022 at 14:20

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