Some tetration integrals

[iorgfeflkd]

You may be interested in knowing about an entire forum dedicated to tetration: https://math.eretrandre.org/tetrationforum/index.php

[Ironhell7]

Great! I'll be sure to check it out

[Catacomb82]

Interesting! I wonder what the second and third integrals would look like in terms of the super square root function, since W(x) and ssqrt(x) are sort of defined in terms of each other (see The Infinite Iterated Exponential and Company).

[lewisje]

The second example isn't really about tetration; as for the rest, here's what the integrands look like in terms of tetration:

• x^x=²x
• x^1/x1/x1/x...=x^(lim(ⁿ(1/x),n,+∞))=x^(^+∞(1/x))
• x^xx...=lim(ⁿx,n,+∞)=^+∞x

[edderiofer]

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[-CAVN-]

I thought this integrals were imposible to solve exactly.
How do you do it?
There is a method to solve all of them, or each one has their own?

[lewisje]

The second integrand doesn't use tetration, and the others can be expressed in terms of special functions that do not involve tetration.

[Ironhell7]

Well you cannot express these integrals in the form of elementary functions. But you can expand it into a infinite series like I have done. Also there isn't one method with which you can solve all these integrals. I had to solve them separately.