Curiosity: there exists the exponential integral?

[Ssnow]

Hi, my question regard the possibility to consider a generalization on the product integral (of type II). The product integral is defined in analogy to the definite integral where instead the limit of a sum there is a limit of a product and, instead the multiplication by $dx$ there is the power of $dx$, in other terms:

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

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:

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

where $\uparrow\uparrow$ is the tetration function ?
Ssnow

 

[anuttarasammyak]

Hi, my question regard the possibility to consider a generalization on the product integral (of type II). The product integral is defined in analogy to the definite integral where instead the limit of a sum there is a limit of a product and, instead the multiplication by dx there is the power of dx, in other terms:

∏abf(x)dx=limn→+∞∏i=1nf(xi)Δxi

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).

Say f(x)>0, it would be
$$e^{\int_a^b \log f(x)dx}$$

I am not sure of the definition of $\uparrow\uparrow c$ where c is not integer. Could you show it to me ?

 

[Ssnow]

You can see this article: https://arxiv.org/pdf/2105.00247.pdf
Ssnow

 

[Ssnow]

Say f(x)>0, it would be
$$e^{\int_a^b \log f(x)dx}$$

This is another form for the product integral ...