some general thoughts - Printable Version

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Hmm,

 I'd like to add two comments. May be, I'm on a wrong track,

 because this requires to change view of things a bit and

 introduces other weaknesses, of which I'm not aware currently.



 The first idea is even a change of a very common view, so

 this is a special slippery path. Let me put it here anyway.



 a) -----------------------------------------------------

 I tend to the opinion, that we should change the view

 of tetration, going away from the assumption, that it

 means



   b, b^b, b^b^b, b^b^b^...



 with the dotted line *at the right* and



   b^^n = b^b^...^b (n times)



 There is already a notation, coined by Andrew Robbins (and I

 actually don't know from which sources this notation stems)

 which supports my other direction of view:

 He defines

   {b,x}^^n :=  b^b^b...^b^x



 The idea is, that x is sort of starting value, and tetration

 recursively appends bases (not exponents), so a coherent

 notation is then for the sequence of partial expressions for

 {b,x}^^inf is then

  x, b^x, b^b^x, ...^b^b^x

 with the dotted line at the left.



 This seems to be an super-artificial difference, but has its

 own impact. It means, for instance, that the evaluation of

 partial expressions is different from (and in opposite direction to)

 the common view of partial evaluation, which was discussed

 in a thread here already.

 The infinite powertower b^b^b^.... cannot take a special

 value "at the top" - since there is no top. And in this view

 it is true, that (using r=sqrt(2)) the infinite expression

 r^r^r^.... approximated by partial expressions converges to 2

 and nothing else, so 2 is the only solution (which also reflects

 the view of Lambert, Euler and the descendent discussion).



 But with this convention we have no tools to include the

 occurence (multiple) "fixpoints" in our formula, and must

 leave this problem open.



 If we redefine tetration as appending bases to a certain

 starting value, as in



   {b,x}^^n :=  b^b^b...^b^x

   {b,x}^^inf := ...^b^b^x



 then also the infinite expression (for n->inf) makes sense

 and is better suited to the concept of evaluation of

 partial expressions. We may then legally insert the various

 fixpoints into x and always have valid expressions.



   {b,x0}^^inf := ...^b^b^x0 = x0

   {b,x1}^^inf := ...^b^b^x1 = x1

  ...

 where x0,x1,... are the fixpoints.

 Note, that this convention is also more coherent with the

 lot of research in iteration-theory and theory of dynamical

 systems, where always an "initial state" is discussed, to

 which then an operator is applied - one time, two times, and

 in generalization even fractional or complex "times".



 *Only* if this notation is accepted, then the following is

 allowed:

  {sqrt(2),2}^^inf = 2

  {sqrt(2),4}^^inf = 4

  ...



 This is *not* allowed (and we even cannot notate a second solution)

 if we write

   sqrt(2)^sqrt(2)^... = ??

 then we can have only one solution and we have no second

 parameter to refer to the different fixpoints.

 (See my posting some monthes ago

   subject: "sqrt(2)^sqrt(2)^sqrt(2)^... = 2 ? or 4 ?"

 where I initiated a discussion of this)



 Another spin-off of a redefinition is then the following.

 Assume

  {b,x}^^h = y

 then, if x is already x = b^b^1 = {b,1}^^2, or more

 general x = {b,z}^^g then we may do a bit of arithmetic

 like



   {b,x}^^h = {b,{b,z}^^g}^^h = {b,z}^^(g+h)



 *iff* the bases for y and x are the same b.

 ANd such an approach complies then easily to the idea

 of a dynamical system, where an initial state (here: z)

 is modified by an operator g and h times.



 Well, *redefining* a common definition is easily a

 crankish behave (see the discussion with A.Plutonium, for

 instance), but in this case I'll take this risk

 and propose a review of our definition.



b) ----------------------------------------------------



 If a) is settled, then we may solve for x in

 {b,x}^^inf = x

 with multiple solutions, assigning branches to the h()-

 function this way.



 To find the various "fixpoints" then can be done by

 a process, which reflects the idea of the newton-approximation,

 for instance for finding the square-root of a number.

 If we want to find x=sqrt(Z) we use an initial guess, say x0,

 compute x1 = (Z/x0 + x0)/2 and iterate. This gives diminuishing

 intervals for the error and approximates the solution to

 arbitray precision.



 With complex initial guesses for a fixpoint in the a)-definition

 of infinite tetration, we can apply this idea equivalently.

 The difference is here, that iterations spiral into the fixpoint

 or spiral away - but seemingly always do spirals.

 Assume an initial value, say x0, and apply an appropriate

 function f(b,x0) and average

    x1 = (f(g,x0) + x0)/2

    x2 = (f(g,x1) + x1)/2

    ...

 then a partial evaluation of such a spiral up to one approximate

 circle gives a center-point near the "mean" of the  spiral

 (or say: the assumed convergence-point).

    y1 = (x1+x2+x3.... xk)

 which can then be a better approximation.

 Then use  x0 = y1 and iterate again up to one complete rotation

 and iterate. This is somehow like throwing a lasso, whose

 diameter tightens radically.



 Depending on the initial guess we may find then the different

 fixpoints for b (well, we may implement the Lambert-W-function

 allowing different branches as well)



 c) This "lasso"-method has another useful consequence: the

 initial spiral may even diverge : it will still circle around a

 center - and this center may then be nearer to the fixpoint

 than the initial guess.

 This may then be seen as an equivalent of Euler-summation of

 alternating divergent series: the angle of "rotation" here is just

 pi, and finding the "center" of a real-valued alternating series

 is then just a special case of this "lasso"-process with two

 steps:

    x1 = something(x0)       (negative value)

    x2 = something(x1)       (positive value again, "circle" closed)

  (which is only a sketch here, since Euler-summation employs

   a binomial-transform of the values)





So a),b) and c) may be a useful framework for the discussion of

fixpoints for complex tetration. I would like to see a better

formal description here (which I cannot supply due to lack of

knowledge) and then a check, whether such description agrees

with the needs of compatibility of the assumptions/results

with the whole surrounding scene of theory and application of

powerseries.



Gottfried Helms