Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let's say $ f(p) $ is a number defined as shown below:

$ \hspace {10 mm}f(p) = {\sqrt {2} ^ {{\sqrt {2} ^ \sqrt {2}} ^ {... \hspace {1 mm} p\hspace {1 mm} times } }} $

What I understand is:
We normally assume, presume or impose a restriction on $p$ that $p$ should be a positive integer.

My question is:
Can we allow $p$ to be any number in general?

      e.g. Can $p$ be irrational? 

If $yes$, how to interpret such a number?
If $no$, why?

Thanks.

share|improve this question
5  
At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of n. Various approaches are mentioned in en.wikipedia.org/wiki/Tetration#Extension_to_real_heights –  Harun Šiljak Jun 23 '12 at 8:12
add comment

2 Answers

up vote 4 down vote accepted

As suggested in meta, I'll turn my comment into an answer (CW).

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of n. Various approaches are mentioned in http://en.wikipedia.org/wiki/Tetration#Extension_to_real_heights .

share|improve this answer
add comment

In addition to the answer of Harun Siljak one should perhaps mention that for the slightly different expression $$ \operatorname{dxp}\small([topexponent],[base],[iterationheight])=\operatorname{dxp}(x,t,h) $$ where $$ \operatorname{dxp}(x,t,0)= x <br>\\\ \operatorname{dxp}(x,t,1)= t^x - 1 <br>\\\ \operatorname{dxp}(x,t,2)= t^{t^x - 1 } - 1 <br>\\\ \cdots $$ there is a solution for real $h$ based on the power series for the exponential function minus the constant term. Usually this is discussed for the function $$ \operatorname{dxp}(x,e,1)= \exp(x) - 1 $$ and fractional or even irrational heights $h$ and a parametrization for the coefficients for $$ \operatorname{dxp}(x,t,1)= \sum_{k=1}^\infty u^k{x^k \over k!} <br>\\\ \operatorname{dxp}(x,t,h)= \sum_{k=1}^\infty \mathcal{P}(u,h,k){x^k \over k!} <br>\\\ \text{where I wrote }u \text{ for } \ln(t) $$ where $\mathcal{P}$ denotes a polynomial in iteration-height, $\ln(t)$ and the series-index $k$.

For series like this and its iterations it is accepted, that the indicated family of iteration heights form a semigroup, where the height-parameter $h$ can be non-integer and can even be complex. This can already be found in L.Comtet's "advanced combinatorics" but also elsewhere.

Unfortunately, although the iterations of dxp() and exp() can be converted into each other (simply by a change of base) for integer heights, this is not uniquely determined for the fractional heights (the reason is, that for the same base in $b^x$ we have multiple bases $t$ in $t^x-1$ and the various $t$ give different results for the same $x$ and height $h$ if $h$ is fractional). Which then leads to the comment in the other answer, that there is not (yet) a commonly accepted interpretation for the noninteger heights in your original problem.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.