I am working with some tetration problems, such as below: $$y = e^{e^x}$$

and I am looking for a concise notation for this. In particular, I would like a way to indicate $n$ iterations of the exponentiation, with the deepest level raised to $x$, rather than $e$.

My initial thoughts were to write the above example as $e^{e^x} = ({^2e})^x = (e\uparrow\uparrow2)^x$. However, since exponential towers must be evaluated from top to bottom, it seems like this is not true.

Is there any other concise notation for this?


  • $\begingroup$ Indeed, have a look into Knuth's superpower notation. $\endgroup$ – Wuestenfux Sep 21 '18 at 12:23
  • $\begingroup$ What about "exp(exp(x))" ? $\endgroup$ – Peter Sep 21 '18 at 12:24
  • $\begingroup$ The $n$-th iterate of $f$ is usually denoted $f^{(n)}$, so I guess $\mathrm{exp}^{(n)}(x)$ would do. $\endgroup$ – Michal Adamaszek Sep 21 '18 at 13:17

(Replacing my earlier comments to make a proper answer)

Suggestion 1: In the tetration forum we have partly used $\exp^{\circ h}_b(x)$ (the little circle indicating function-composition instead of powers or instead of derivation) for the general iteration $x \to b^x$ to the iteration-(h)eight $h$ of the exponentialtower. I myself use sometimes $\text{T}^{\circ h}_b(x)$ for shortness and $\text{U}^{\circ h}_b(x)$ for the decremented exponentiation $x \to b^x−1 $.

Suggestion 2: In many articles I've also seen the simple solution to use the index-notation. So $z_0$ for the initial value , $z_1=b^{z_0}$ then $z_h=b^{z_{h−1}}$ for the $h$'th iteration (exponentialtower of (h)eight $h$) and $z_\infty$ if that limit exists. In articles the base $b$ is mostly a fixed parameter over a lot of formulae and algebraic derivations so I'd prefer such a notation which allows to omit this reference to $b$ to reduce redundancy in notation. (I find this unbeatable concise - unfortunately the indexing-notation indicates many things in math so I use this only when I'm well sure it is not obfuscating my line of discussion/derivation/definition)


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