It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any simple form of the inverse known?
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2Probably most people wouldn't consider expressions that involve the Lambert W function closed-form. – Travis Willse Aug 30 '14 at 18:40
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@Travis: Based on what they do not considerate as a closed form? – Mhenni Benghorbal Aug 30 '14 at 18:47
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Short answer no. Long answer: Some very long algebraic arguments using W(x). – Ali Caglayan Aug 30 '14 at 18:49
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By wikipedia's definition; In mathematics, a closed-form expression is a mathematical expression that can be expressed using a finite number of symbols. These symbols include constants, variables, certain well-known operations (e.g., + − × ÷), and certain well-known functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions). Lambert-W function isn't an elementary function so the answer is no. – UserX Aug 30 '14 at 19:03
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@ClaudeLeibovici $y=(x^x)^x$ is probably not equal to what the OP is asking. – Simply Beautiful Art Dec 19 '15 at 18:49
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Perhaps this link will help: http://math.stackexchange.com/questions/1583907/solution-to-eex-x-and-other-applications-of-iterated-functions – Simply Beautiful Art Dec 21 '15 at 00:40
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@Ali Caglayan LOL, you can express it using Lambert function? Please, make an answer! – Anixx Nov 26 '16 at 21:00
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@Travis LOL, you can express it using Lambert function? Please, make an answer! – Anixx Nov 26 '16 at 21:01
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@Anixx I doubt that it is possible to do so, at least not without invoking additional special functions. – Travis Willse Nov 28 '16 at 10:49
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I have discussed this a bit, but only got a power series, not a closed form. see http://go.helms-net.de/math/tetdocs/Wexzal_Superroot.pdf for the initial essay, then on my announcement in the "Tetrationforum" there came up a discussion http://math.eretrandre.org/tetrationforum/showthread.php?tid=1033 (superroot discussion) http://math.eretrandre.org/tetrationforum/showthread.php?tid=1033&pid=8137#pid8137 a concurring approach http://math.eretrandre.org/tetrationforum/attachment.php?aid=1216 the paper (possibly not directly downloadable) – Gottfried Helms Dec 04 '16 at 07:14
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The closest possible solution comes from noting $x^{x^x}=$ $^3x$.$$^3x=y$$$$x=\sqrt[3]{y}_s$$Also known as the super-cube root. Similar solution applies to higher towers.
Unlike $\sqrt{y}_s$, other super-roots are not definable in terms of known functions.
Simply Beautiful Art
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