- #1

Ssnow

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[tex] \prod_{a}^{b}f(x)^{dx}\,=\, \lim_{n\rightarrow +\infty} \prod_{i=1}^{n}f(x_{i})^{\Delta x_{i}}[/tex]

considering a reasonable partition of the interval ##(a,b)##, (you can find references on wiki with a lot of details for product integral of type II or others on the web).

The question is, there is a generalization of this by "analogy'' using exponentiation and tetration function (extended to real heights), this will be formally:

[tex] EXP_{a}^{b}f(x)\uparrow \uparrow dx\,=\, \lim_{n\rightarrow +\infty} \left(f(x_{i})\uparrow \uparrow\Delta x_{i}\right)^{\left(f(x_{i})\uparrow \uparrow\Delta x_{i}\right)^{(\cdots)}}\ \ (n-times)[/tex]

where ##\uparrow\uparrow## is the tetration function ?

Ssnow