I've recently seen in my studies tetration, or the next operation in the addition, multiplication, exponentation... series. I've also heard much discussion about how to extend this operation to "heights" (that is, the number of times to which the exponentation is repeated) other than the natural numbers. Now given that I've had not much higher math experience (pretty not much farther than real analysis) and I might very well be oversimplifying things, but my question is: why can't we define fractional 'heights' like we do with exponents?

If $x$ is any base, then $\displaystyle x^{1\over 2}$ is ${\rm sqrt}(x)$. Similarly, why can't $\displaystyle {}^{1\over 2}x$ be its inverse, ${\rm ssrt}(x)$ (I will use ${\rm ssrt}(x)$ as its inverse and ${\rm scrt}(x)$ for the inverse of tetration with height $3$). It also works for expressions like $5\over 3$: ${}^{5\over 3}x = {}^{5}{\rm scrt}(x)$. As for irrational numbers, well I guess that's another story, just like it is for exponents.

  • $\begingroup$ Welcome to MSE! It really helps readability to format using MathJax (see FAQ). Regards $\endgroup$ – Amzoti May 25 '13 at 13:50

If you write an operator-notation for your function $f(x)$ say the operator $E$ for the exponential-function, such that $$ \exp(x) = E \circ x \qquad \qquad \text{and} \\ \exp(\exp(x)) = E \circ E \circ x \qquad \qquad \text{ and } \ldots $$ then tetration of a certain height $h$ must be written as $$ \exp^{\circ h}(x) = E^h \circ x $$ and the question is, whether or how we could invent/understand an operator and even a power of an operator.

Now one idea to implement an operator for analytic functions (which have a power series with nonzero range of convergence) there is the concept of "Carleman"-matrices (see wikipedia) which is construed for the manipulation of the coefficients of the (formal) power series.
With that you need thus the concept of powers and even of fractional powers of matrices.
For instance for rational/linear functions $f(x)$ the Carleman-matrix is nearly trivial: a 2x2-matrix, of which you can then define a fractional power using diagonalization or logarithmization very simply. And this can even be understood and done by any amateur.

However, the inverse function, say to find the height of a tetrations needs to find the exponent for the Carleman-matrix which would have iterated $x$ to $x_h$ (the h'th iteration of the function). Here the logarithm of a matrix becomes involved...

For general functions the matrices become of infinite size and it is non-trivial to define meaningfully fractional powers of such matrices. Accepted solutions are so far when the infinite Carleman-matrix is lower triangular (which means the powerseries of the function $f(x)$ has no constant term and has thus a fixpoint at zero).

For the exponentiating this is exceptionally difficult because also the inverse (the logarithm) is multivalued and the exponential-function has no real fixpoint.


Another perspective is that of the "Schröder-function" which can intuitively be seen as an analogon to the log/exp-function for fractional iteration of the multiplication. If we ask for the fractional height of iterated multiplication with a certain base $b$, say $$f: x_1 = b \cdot x_0 \qquad \qquad \text{ and } \\ f°^2 : x_2 = b \cdot b \cdot x_0 \qquad \qquad \text{ and } \\ f°^h : x_h = \underset{ h \text{-times}}{\underbrace{b \cdot b \cdot b \cdot \ldots b \cdot} } x_0$$ then $h$ is the "height" of iterated multiplication. To find the iterated multiplication with a fractional height $ x_h = f°^h(x) = b^h \cdot x_0$ we use the pair of log/exp-function in the following way. First there is a constant $u_b$ defined with the "base" $b$ (in the context of multiplication ) such that $u_b=\log(b)$. Then we write: $$ x_h = \log°^{-1}( u_b \cdot h + \log(x_0)) $$
$ \qquad \qquad \qquad $ (where of course $\log°^{-1}(x)$ means $\exp(x)$)

For the question of the inverse: determine, which height $h$ is required to iterate from $x_0$ to $x_h$ by multiplication with base $b$ ? we employ the $\log$ for both $x_0,x_h$ and thir height-difference is then linear in h: $$ \operatorname{hgh} : h = {\log(x_h) - \log(x_0) \over u_b }$$

Similarly there exists the "Schröder-function" which is used in a roughly analoguous way. For a base $b$ in connection with the function-iteration there is some constant $u_b$ defined, which after transformation of the initial value $x_0$ the h'th power of $u$ leads to the $h$'th iterate. We have for the h'th iteration $$ x_h = \operatorname{Schröder}°^{-1} ( u_b^h \cdot \operatorname{Schröder}(x_0)) $$

(we might say, the $\log$ is the Schröder-function for the iteration of multiplication, while the Schröder-function for the iterated exponentiation has not yet a name).

You can read more on Schröder-functions in the wikipedia; possibly this concept is even nearer to the focus of your question than that of the Carleman-matrices although I found the Carleman-matrices easier to understand (even I arrived at this incidentally by my own fiddling with powers of matrices...)


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