There are ways to express the function $$f(x)=x^{x^{x^{x^{x^{x^\cdots}}}}}$$ with $f(x)=\dfrac{W(-\ln(x))}{-\ln(x)} $ for other function like this; $$g(x) = x^{-x^{x^{-x^{x^{-x^{x^\cdots}}}}}} $$ I don't know how to express in a simple function, but I can put it like. $$ \exp\left({\frac{W(-\ln(y^y))}{y}}\right) = x$$ But for something that doesn't repeat like. $$h(x)= x^{-x^{-x^{x^{x^{x^{-x^{-x^{-x^{-x^\cdots}}}}}}}}} $$ How could I write it as an implicit function? Or find the value of $h(e)$?

  • $\begingroup$ Is defined as -(x^x) and the next part as x^-(x^(-x)). I thought it looked a bit cumbersome with all the parentheses $\endgroup$ – jose valenzuela Dec 22 '19 at 17:15
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    $\begingroup$ How is $h(x)$ defined? Are you asking about some $h(x)=x\text{^}(a_1x)\text{^}(a_2x)\text{^}(a_3x)\text{^}\dots$ such that $a_k\in\{-1,+1\}$ defines some non-periodic ("non-repeating") binary sequence? $\endgroup$ – Vepir Dec 22 '19 at 18:31
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    $\begingroup$ I'm not sure how would I define the sequence, but it would be like 0, 1, 1, 0, 0, 0, 1, 1, 1, 1 . . . $\endgroup$ – jose valenzuela Dec 22 '19 at 18:45

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