Tetration Mathemat iiicalcalcal Gottfried Helms - Univ Kassel 01 - 2008 Miniatures Tetration by matrix-operations Preview! missing references and indexing of equations Abstract: An introduction into my version of the extension of discrete to continuous itera- tion is given. Tetration is expressed in terms of powerseries, and iteration as action on formal powerseries (exponentialseries). To be able to repeat the basic operation of expo- nentiation in this framework, the concept of use of matrices is introduced; the matrices which transform one powerseries in x into another of f(x) are called "operators" here. This text-version is only a preview; only the introduction and the part about "naive tetra- tion" (using finitely truncated matrices as sufficient with basic matrix-operations) is pre- sent – the part about the full "analytical approach" is not yet ready. Version: 0804-04 0803-30 Contents 1. "Iterated exponentiation" ( IE or T(x)) and decremented IE (DIE or U(x)).................... 2 1.1. Introduction.............................................................................................................. 2 1.2. The exponential-series and its iterate....................................................................... 5 1.3. Fractional iterates..................................................................................................... 5 1.4. The matrix-approach................................................................................................ 6 1.5. Matrix-notations and definitions.............................................................................. 8 2. The naive matrix-approach............................................................................................ 10 2.1. Example computation of x {b} h with integer height h >=0 ................................. 10 2.2. Negative height...................................................................................................... 10 2.3. Acceleration of convergence - Euler-summation .................................................. 11 2.4. Fractional powers: matrix-logarithm or diagonalization ....................................... 12 2.5. Sums and series of powertowers (here: alternating series).................................... 13 2.6. naive U-tetration (decremented iterated iteration)................................................. 16 2.7. Improvement of naive T-tetration by fixpoint-shift............................................... 16 2.8. Conclusion ............................................................................................................. 17 3. An analytical matrix-approach...................................................................................... 18 3.1. Intro........................................................................................................................ 18 4. citations/snippets............................................................................................................ 20 5. Appendix ........................................................................................................................ 21 5.1. Relation between Bell-matrix B[U] and Ut-matrix................................................ 21 6. References and online-resources ................................................................................... 22 Project-Index http://go.helms-net.de/math/binomial/index Intro/notation http://go.helms-net.de/math/binomial/intro.pdf Tetration using matrices S. -2- 1. "Iterated exponentiation" ( IE or T(x)) and decremented IE (DIE or U(x)) 1.1. Introduction In the following I'll discuss two different types of tetration, x x, b x, b b ,... - T-tetration, "iterated exponentiation" footnote a (IE) x x, t x-1,t t -1-1, ... - U-tetration, "decremented itereated exponentiation" (DIE) I shall call one such term bbx " powertower ", which has the " base " b, the " height " (or " iteration- parameter ") h (number of b's) and the " top-exponent " (or " initial value ") x. Note, that for exponentiation I also use the caret-symbol " ^" like b^b^x with evaluation from right be- cause of difficult typesetting if the height is more than one or two. Also, from common use in the tetration-discussion where no special parameter x is assumed, I adapt the notation b^^h for bb...b with b's h times occuring (this is using x=1 in my notation) for shortness. T-Tetration : let’s define the first version in functional notation x Tb(x) = b °0 °h °h-1 (footnote b) Tb (x)=x Tb (x) = T b (T b(x)) The height-parameter h may be assumed as integer in the beginning and complex in the following chap- ters. °h There is no restriction on x; however, if it is x=1 , then we deal with the classical tetration, then Tb (1) = b^^h . °h In the more current notation of member of tetration forum Tb (1)= b[4] h while I announce the ascii- version for my subject of discussion °h T b (x)= x {4,b} h , which is immediately concatenable x {4,b} h 1 {4,b} h 2 = x {4,b}( h 1 + h2 ) , and the more convenient notation, since the operator-number " 4" is assumed here everywhere: °h T b (x)= x {4,b} h = x {b} h U-tetration define the second version as x Ut(x)=t – 1 °0 °h °h-1 Ut (x)= x Ut (x) = U t (U t(x)) proposing the ascii-version for this text: °h Ut (x)= x {4_,t} h = x {_ t} h The same restrictions are valid for t,h,x as before. If the subscript t is omitted, it is meant as a general remark or the current base is assumed. I use the letter t instead of b for the base-parameter, since U- and T-tetration are nearly related with a certain pair of bases, so it may be useful, to distinguish the letters from the beginning. a see [TF08 1], "Notations and Opinions", Andrew Robbins in Tetration-forum; [Wiki:TE] "Tetration" in Wikipedia, or "Tetra- tion in Context" [AR07 09], same author b Note the recursive definition of the iteration. Sometimes I see ([Wiki:TE] for instance) this written differently wrt the height/iterator-parameter: °h °h-1 Tb (x) = T b(T b (x)) which I do not propose. The problem is here, that for a limit in the infinite recursion, where h-1 is still infinite, we never had a starting operation except the evaluation of the –again– infinite partial powertower in advance. Tetration Mathematical Miniatures Tetration using matrices S. -3- Base-parameter and domain The base-parameter b deserves a specific consideration. In the analytical discussion we'll make frequent use of the following relation of symbols: b = t 1/t t=exp(u) So in fact, concerning the domain for the base-parameter, I tend to use u as primary variable, which may assume any complex value. Depending on this, t is t=exp(u) and the range for t is thus C \ 0. Since t cannot be zero, this definition also prevents the discussion of b as b=0 1/0 and also b can assume the value 0 only as a limit, if at all. So actually this formulates a map u → exp(u/exp(u)) which is C → C \ 0 With the notion of "iterated exponentials" or "decremented iterated exponentials" there are two tiny, but significant differences to the common concept of tetration, or powertower. First, the additional parameter x is not known in the usual idea of tetration and powertower (it may be assumed to be x=1 to model tetration). In T- and U-tetration it is seen as a starting value for the re- peated operation. The terms "iterated exponentiation" for the T-operation and "decremented iterated exponentiation" for the U-tetration were proposed by Andrew Robbins [TF 1]; I’ll follow that in princi- ple, but for simpliness I misuse the more common term of tetration as a generalism. Second, the order of operation is not in contrast to the indexing of the parameters here. For instance in D.F.Barrow in [BR36] the general iterated exponentiation is indexed as a0^a 1^...^a n although the evaluation is right associative, thus beginning with an. D.F. Barrow, 1936 This does not make a difference to the usual concept of powertowers/tetration in the finite height. But it is significant when want to deal formally with infinite height. If the evaluation of powertowers for the infinite case is discussed in the common way, then it is thought in terms of partial evaluation b, b^b, b^b^b, b^b^b^...=b, b^^2,b^^3,b^^4, ... b^^inf and then obviously in the infinite case there cannot be assumed a parameter "in infinity" different from b. This way already Euler studied this operation and found a range of convergence for b, namely e-e <=b<=e 1/e . Let’s call this range the „ Euler-range “. However, the observation, that for b=sqrt(2) the sequential partial evaluations b^2 = 2, b^b^2 = 2, ..or: 2 {b} h = 2 for all h>0 b^4 =4, b^b^4 = 4, ....or: 4 {b} h = 4 for all h>0 Tetration Mathematical Miniatures Tetration using matrices S. -4- are constant with two different „top-parameters“ t lets us perplexed. Euler discussed this also (although it is inconsistent with the said partial evaluation) and concluded 1/2 t1=2 b^2 = 2 b=2 1/4 1/2 t2=4 b^4 = 4 b=4 = 2 2 {4,b} inf = 2 and 4 {4,b} inf = 4 ==> b=sqrt(2) but finally the powertower/tetration-concept didn’t adapt the required redefinition of indexing. In the definitions here the above considerations read without contradictions in the infinite case as x = x {b} 0, b^x = x {b} 1 b^b^x=x {b} 2, b^...^b^b^x = x {b} h, // h-times b repeated ...^b^b^x = x {b} inf, // infinitely many b’s assumed where the dots occur at the left side of the expression. Then setting x=0 , x=1 , x=b or general x= 1{b} h allows to write the infinite case for two different t, where b=sqrt(2) 1/t 1 t1 {b} inf = t 1 => b = t 1 1/t 2 t2 {b} inf = t 2 => b = t 2 allowing statements about the infinitely iterated case, beginning with an arbitrary starting-parameter t, which may assume the value of one of multiple possible fixpoints. So, if bt = t, b^b^t = b^t = t, ... this doesn’t change, how many b’s we append to the left. Then t is called a fixpoint for b. The concept of fixpoints is essential for non-integer tetration, but we shall discuss this later. Also it should be men- tioned here, that there are not only two, but infinitely many fixpoints for each base b, if the complex domain is considered for b and t.