Curiosity: there exists the exponential integral?

• I
• Ssnow
In summary, 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:
Ssnow
Gold Member
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

Ssnow said:
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 ?

Last edited:
anuttarasammyak
anuttarasammyak said:
Say f(x)>0, it would be
$$e^{\int_a^b \log f(x)dx}$$
This is another form for the product integral ...

1. What is the exponential integral?

The exponential integral, also known as the Ei function, is a mathematical function that appears in many areas of science and engineering. It is defined as the integral of the exponential function from 0 to infinity.

2. What is the significance of the exponential integral?

The exponential integral is important in many applications, including physics, chemistry, and statistics. It is used to solve differential equations, calculate probabilities, and model physical phenomena.

3. How is the exponential integral calculated?

The exponential integral cannot be expressed in terms of elementary functions, so it is typically calculated using numerical methods or special functions such as the incomplete gamma function. There are also tables and software programs available for calculating the exponential integral.

4. Can the exponential integral be negative?

Yes, the exponential integral can have negative values for certain input parameters. In general, it is a complex-valued function that can take on both positive and negative values.

5. What are some real-world examples of the exponential integral?

The exponential integral has many real-world applications, such as calculating the electric potential of a point charge, modeling the decay of radioactive materials, and determining the probability of extreme events in statistics. It is also used in the study of black holes and the behavior of light in gravitational fields.

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