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This question already has an answer here:

Tetration is defined for all $n\in \Bbb{N}$ by $$ {^1}a = a \\ {^{n+1}}a = a^{\left({^n}a\right)} $$ Thus ${^3}a$ means $a^{a^a}$.

Here $a$ could be any real (or indeed even complex) value, but only integer $n$ in the tetration is defined by this definition.

But I've seen questions here dealing with real valued tetration (${^x}a$ for real $x$) and even one dealing with complex tetration
(${^z}a$ for complex $z$).

let's take one step at a time: How does one define tetration with a rational hyperpower $${^q}a : q \in \Bbb{Q}$$ (From there, the move to real tetration is the same sort of "easy" as defining real exponentiation.)

And what does ${^z}a$ mean for complex $z$?

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marked as duplicate by Matthew Towers, Frunobulax, Harish Chandra Rajpoot, hardmath, Pierre-Guy Plamondon Feb 22 '16 at 12:39

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Another good question, why would you define such a thing? $\endgroup$ – bof Feb 22 '16 at 7:04
  • $\begingroup$ @bof: Lars Kindermann had written his thesis about the fractional iteration of the exponentiation in the context of an industrial process of producing sheets of steel in a set of consecutive steps by identical machines. However he approched/solved this using neural networks. (The thesis (in German, having lots of interesting references) is online) $\endgroup$ – Gottfried Helms Feb 22 '16 at 8:31
  • $\begingroup$ @bof Fundamental physics makes use of exponentiation, determining whether the next arithmetic operator tetration has or doesn't have a role in fundamental physics is interesting. $\endgroup$ – Daniel Geisler Feb 23 '16 at 5:16
  • $\begingroup$ You can take a look at mathoverflow.net/q/259278/9550 $\endgroup$ – Vladimir Reshetnikov Jan 11 '17 at 0:20
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The most intuitive construction for this is to have a infinitely-sized matrix A which performs $$ V(x) \cdot A = V(a^x) \qquad \qquad \text{ where } V(x)=[1,x,x^2,x^3,...] $$ Such a matrix A is called "Carleman-matrix" (see wikipedia). The idea is, that then $$ V(x) \cdot A = V(a^x) = V(\exp_a(x))\\ V(a^x) \cdot A = V(a^{a^x}) = V(\exp_a^{[2]}(x)) \\ \ldots $$ or $$ V(x) \cdot A^h = V(\exp_a^{[h]}(x)) $$ thus requiring the h'th power of the matrix $A$.
Approximations to this can be made by finite truncations of A - so let's in the following denote such finite truncations with the hat-symbol: $ \hat A$ .

It is so far not difficult to implement arbitrary precision to this if the base $a$ is in the range $\small 1 < a < \exp(\exp(-1))$ with finite matrices $\hat A$ - however, who needed such a construction for the integer height tetration?

But from here one idea is apparently straightforward: we need fractional powers of A to have fractional heights of tetration. For finite-size matrices one can define fractional powers using diagonalization and take the scalar elements of the diagonal matrix to the (fractional) h'th power. So finite truncations of A and its diagonalization gives an approximation.
Let the diagonalization of the truncated matrix $ A$ be written as $$\hat A = \hat M \cdot \hat D \cdot \hat M \, ^{-1}$$ and for notational convenience let's write $\small \hat W$ for $ \small \hat M \, ^{-1}$ then $$ \hat A^h = \hat M \cdot \hat D \, ^h \cdot \hat W = \hat M \cdot \text{diag}( (d_{k,k})^h) \cdot \hat W $$ and -in principle- $$ \hat V(x) \cdot \hat A \,^h \approx \hat V(\exp_a^{[h]}(x))$$ Unfortunately the truncation of $A$ spoils its Carleman-characteristic, and the best approximation is to use the second column only $$ \hat V(x) \cdot \hat M[,1] = \hat \sigma_a(x) \\ \hat V(\hat \sigma_a(x)) \cdot \hat D \ ^h \cdot \hat W[,1] \approx \exp_a^{[h]}(x)$$

See a worked example in Pari/GP in my answer at the earlier question (your question here seems to get marked as a duplicate of that earlier one)


Remark: the last procedere mimicks in principle the method of Ernst Schröder (the $ \hat \sigma _a(x)$ represents here the "Schröder-function") and can be accurate to the -in my opinion- currently best concept for the real tetration on the real axis by H. Kneser (implemented by members of the Tetrationforum) to 10 to 12 or more digits using matrix size of 32x32 . But even size 16x16 allows reasonable approximation to say 6 or 8 digits (don't know exactly from the top of my mind)
Another related concept, immediately implementable for bases a in the mentioned interval is to use Carleman-matrices for the $\exp_a()$-function but where the power series are recentered at an attractive fixpoint. Then the Carleman-matrices become triangular and diagonalization is simpler (and the results are slightly different). Also various other concepts have been discussed but are mostly far more complicated.

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