Solve Integral: x^(1/x) to Power of x^(1/x)

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Discussion Overview

The discussion revolves around the integral of the function defined as [x^(1/x)]^[x^(1/x)] repeated infinitely. Participants explore the validity and potential methods for integrating this function, questioning its well-defined nature and integrability.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how to approach the integral, indicating a lack of prior work on the problem.
  • Another participant questions whether the function is well-defined or integrable, suggesting that the colleague may be joking.
  • A participant introduces the concept of a power tower, noting that the function's definition may depend on the limiting process used to define it.
  • There is skepticism about the integral's legitimacy, with one participant suggesting it may not be a serious problem.
  • A participant asks if the function can be resolved into a definite form that is integrable, indicating a desire for further exploration.

Areas of Agreement / Disagreement

Participants generally express skepticism about the well-defined nature of the function and its integrability, with no consensus on whether it can be resolved into a definite integrable form.

Contextual Notes

Participants note the potential dependence on definitions and the need for a limiting process to define the function properly. There are unresolved questions regarding the mathematical steps necessary to approach the integral.

Who May Find This Useful

Readers interested in advanced calculus, integral theory, or mathematical reasoning related to undefined functions may find this discussion relevant.

latyph
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how do i get to solve this integral,i have no idea whatsoever so no one should expect what i have done.it was presented to me by a colleague
[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]^[x^(1/x)]...
 
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Do you mean the integral of that function?
Do you have any reason to believe that function is well-defined, much less integrable?
Does your colleague have a penchant for pulling your leg?

A couple of points: since [x^a]^b= x^{ab}, [x^{\frac{1}{x}}]^{x^{\frac{1}{x}}}= x^{\frac{2}{x}}. In general then, that "stack" of x^{\frac{1}{x}}, n times, is the same as x^{\frac{n}{x}} and I see no reason to think that expanding it to infinity will give a function.
 
Power Tower

This is not well defined, it seems. Consider that e^{x^{2}} is not the same as \left(e^{x}\right)^{2}=e^{2x}. The one way to look at the given function is as a power tower (e.g. x^{x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}}},) see http://mathworld.wolfram.com/PowerTower.html for a reference; and another way is as HallsofIvy pointed out. I suppose it would depend on how, that is, by what limiting process, the given integrand is being defined. You might try defining the function as a limit of a sequence of functions, perhaps you can use the Lebesgue's Dominated Convergence Theorem to show convergence of the integral (supposing it's a definite one).
 
i get the feeling that's a bull**** integral that the 'colleague' gave out...
 
Why? Do you suppose it is homework?
 
But Cant The Function Be Resolved To A Definite One That Can Be Integrable
 

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