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

[latyph]

--------------------------------------------------------------------------------
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)]...

 

[HallsofIvy]

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.

 

[benorin]

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).

 

[fourier jr]

i get the feeling that's a bull**** integral that the 'colleague' gave out...

 

[benorin]

Why? Do you suppose it is homework?

 

[latyph]

But Cant The Function Be Resolved To A Definite One That Can Be Integrable