(This article is a light introduction. For more austere results, consult Manuscript #2).

We repeat here the main problem for completeness, so that the reader has the main objective in mind.

**Main Problem**^{[1]}

If F(x,n) = x^{xx...} iterated n-1 times, is there a
definition of a *continuous* function identical to F, not limited to integral
values of n, and valid for larger values of x (Greg Kavalec in sci.math)?.

Abstract

The problem has been known under the name *continuous extension of the hyper4
operator*, *continuous extension of the Ackermann function*, *continuous
extension of the hyperexponentiation functions*, and under various different
names.

The main difficulty of the problem lies with the non-associativity of
hyperexponents. The problem of non-associativity of hyperexponents has been
investigated thoroughly, in the references section, with those articles that deal with
the equation x^{y} = y^{x}.

Readers interested in several preliminary results, should consult the references and
article 3, which introduces the notion of the
*hyperroot/tetraroot*, with an appropriate analysis and corresponding code to find
those roots using Lambert's W function.

The main problem showing up after one has defined ^{1/n}m to be the n-th
order hyperroot of the real number x, is that there are two "natural" ways to define
^{m/n}x, either as ^{m}(^{1/n}x) or
^{1/n}(^{m}x). If there is to be any hope of consistency, those two
definitions should produce identical results.

Unfortunately, it is easy to see that the above definitions give in general
different results, unless m=n. Therefore, such a path that hopes to define
^{r}x for rational r, is naturally doomed to failure^{[2]}.

Additional hints that doom this path can be easily gotten when one considers
^{m/n}x and ^{m'/n'}x, with m/n = m'/n'. There are problems there too,
although in some of the references suggestions are given for possible definitions
similar to lim_{k->+∞}^{(km/(kn))}x.

In this presentation we examine an alternate construction, which does not suffer
from an inconsistency definition-wise and the final function acquired via the
definition has the property that it preserves *continuously* the behavior of all
the "known" hyperexponentials ^{n}x for natural n.

Recall our general exponential function again, for completeness here, f(z,w,n) = {z^{w}, iff n=1,
z^{f(z,w,n-1)} iff n>1} (1)

We will use the f above, slightly modified to aid us in the extension. Let's then define our new F as follows:

F(z,w,n) = {z^{w}, iff n in {0,1}, z^{F(z,w,n-1)} iff n>1}
(2)

Let [x] = Integer Part Of x. Define for rational r > 0:

^{r}x={F(x, r, 0)=x^{r} iff [r] = 0, F(x, x^{r-[r]}, [r])
iff [r] > 0}

We need the following lemma to make the extension:

Lemma:

If {r_{k}}, k in N, is a Cauchy sequence of rationals with
lim_{k->+∞}r_{k} = y in R^{+}-N, {[r_{k}]} is
also a Cauchy sequence of rationals with lim_{k->+∞}[r_{k}] =
[y].

Proof:

Given any ε, there is a k_{0} in N, such that for all k >
k_{0}: |r_{k} - y| < ε. In particular pick a k_{0},
that guarantees, for all k > k_{0}: |r_{k} - y| < ε =
min{|y - [y]|, |[y] + 1 - y|}. It is clear then that for all k > k_{0}:
[r_{k}] = [y], therefore, |[r_{k}] - [y]| = 0 < ε and the
result follows.

The reader may wonder here why the author excluded the naturals from the above
Lemma. If {r_{k}} is *any* Cauchy sequence converging to n, the Lemma may
fail, because if we approach n from the left via a Cauchy sequence of rationals
{r_{k}}, such that r_{k} < n, for all k: |[r_{k}]-n| =
|n-1-n| = 1 > ε for *all* k. So one should be careful to pick a Cauchy
sequence that approaches n from the right. Note however that in this case a Deus ex
Machina saves us: n is in Q, and we are covered by the definition for rational r =
n.

Corollary:

If {r_{k}}, k in N, is a Cauchy sequence of rationals with
lim_{k->+∞}r_{k} = y in R^{+}-N, {r_{k} -
[r_{k}]}, k in N, is also a Cauchy sequence of rationals with
lim_{k->+∞}(r_{k} - [r_{k}]) = y - [y].

Proof:

The sum of two fundamental sequences is again fundamental.

We can now naturally extend the definition unambiguously. Let {r_{k}}, k in
N, be a Cauchy sequence of rational numbers, with y =
lim_{k->+∞}r_{k}, y > 0, y in R^{+}.

^{y}x = {lim_{k->+∞}F(x, r_{k}, 0) =
lim_{k->+∞}x^{rk} iff [y] = 0,
lim_{k->+∞}F(x, x^{rk-[rk]},
[r_{k}]) iff [y] > 0}^{[3]}.

Will these work? We hope so. Let's see why.

Claim:

The function ^{x}y is continuous for all x > 0 for fixed y > 0.

Lemma #1:

Given y > 0, F(y, x, n) is continuous for all x >= 0, n > 0, n in N.

Proof:

By induction on n. y^{x} = F(y, x, 1) = e^{log(y)*x} is continuous
for all x >= 0. But F(y, x, k+1) = y^{F(y, x, k)}, the last being continuous
as the composition of y^{x}, which is continuous from the n=1 step and F(y, x,
k) which is continuous by the inductive step.

Some Observations:

Fixing y > 0, it is clear that if x is away from integral values (i.e. if x is in
R^{+}-N), we don't have a problem, since there, x - [x] is continuous,
therefore small changes in x will intuitively result in small changes for
^{x}y, no matter how high the tower is.

The only "suspicious" points where continuity may actually fail, are the natural
numbers (because x - [x] is discontinuous there). The natural numbers are in fact,
*the points of transition*, where F "acquires" additional exponents.

Let's see what happens:

Lemma #2:

For fixed y > 0, ^{x}y is continuous for all x: x in N.

Proof:

By induction on n. For n = 1, we get:

|^{1+dx}y - ^{1}y| =

|F(y, y^{1+dx-[1+dx]}, [1+dx]) - F(y, 1, 1)| =

|F(y, y^{dx}, 1) - F(y, 1, 1)| =

|y^{ydx} - y^{y0}| < ε, (by Lemma #1,
y^{yx} is continuous at 0)

On the other hand:

|^{1-dx}y - ^{1}y| =

|F(y, 1-dx, 0) - F(y, 1, 1)| =

|y^{1-dx} - y^{1}| < ε, (by Lemma #1, y^{x} is
continuous at 1)

Assume now that ^{x}y is continuous at x = k. Then:

|^{k+1+dx}y - ^{k+1}y| =

|F(y, y^{k+1+dx-[k+1+dx]}, [k+1+dx] - F(y, 1, k+1)| =

|F(y, y^{dx}, k+1) - F(y, 1, k+1)| =

|F(y, y^{ydx}, k) - F(y, y^{y0}, k)| <
ε, (by the inductive step, composition and Lemma #1)

|^{k+1-dx}y - ^{k+1}y| =

|F(y, y^{k+1-dx-[k+1-dx]}, [k+1-dx] - F(y, 1, k+1)| =

|F(y, y^{1-dx}, k) - F(y, 1, k+1)| =

|F(y, y^{1-dx}, k) - F(y, y^{1}, k)| < ε, (by the inductive
step, composition and Lemma #1)

And Lemma #2 is proved. Lemma #2 along with continuity of ^{x}y at the
non-transitional points: {x: x in R^{+} - N} (which follows trivially from
Lemma #1) proves the main claim.

Let us now see how this function behaves with Maple:

> f_N:=proc(z,w,n)

> option remember;

> if n=0 or n=1 then z^w;

> else z^f_N(z,w,n-1);

> fi;

> end:

And the function ^{r}x:

> f_Q:=proc(z,r)

> local fracpart,intpart;

> intpart:=floor(r);

> fracpart:=r-intpart;

> if intpart>0 then

> f_N(z,z^fracpart,intpart);

> else #intpart=0

> f_N(z,fracpart,0);

> fi;

> end:

Let us now verify some of the more elaborate properties of the hyperexponential.

First, we know that lim_{n->+∞}^{n}x =
e^{-W(-log(x))}, whenever x is [(1/e)^{e}, e^{(1/e)}]. Perhaps
we can then see some similar behavior on our ^{r}x (green line below is
e^{-W(-log(x))}).

At the left bound of convergence, (1/e)^{e}:

> plot({'f_Q(exp(-1)^exp(1),r)', exp(-1)},'r'=0...30);

At the right bound of convergence, e^{(1/e)}:

> plot({'f_Q(exp(exp(-1)),r)', exp(1)},'r'=0...30);

Right of the right convergence bound:

> plot({'f_Q(1.9,r)'},'r'=0...3.5);

Left of the left convergence bound:

> plot({'f_Q(0.02534,r)'},'r'=0...6.5);

Note that in all cases, ^{r}x is continuous at the transitional points r = n
for fixed x. Note also how the function behaves left of (1/e)^{e} where it is
known that ^{n}x is a two-cycle. On the other hand we have also:

Lemma #3:

For fixed y > 0, ^{y}x is continuous for all x in R.

Sketch of Proof:

This amounts to showing that for fixed y, the function: F(x, x^{y-[y]}, [y])
is continuous. But when y is fixed, then so is [y] = n and so is q = y-[y], and in this
case F(x, x^{y-[y]}, [y]) = x^{x...xq}
(n + 1 x's). This follows easily by using induction on n, the fact that x^{y}
is continuous for fixed y and from composition of continuous functions, with a similar
argument of that in the proof of lemma #1 (see ^{[3]} as above).

Now, we should be able to see the desired extension as a nice continuous
interpolation between the graphs of the functions ^{n}x for integral n. And
indeed:

> plot({seq(f_Q(x,2+n/10),n=1..10)},'x'=0..2);

The more interesting initial and final segments of the above graph (to view those
segments, change the function bounds to x=0..1 and x=1..2 in the Maple plots
above)^{[4]}.

To watch the interpolation as an animation:

> display([seq(plot('f_Q(x,2+n/10)','x'=0..2),n=1..10)],insequence=true);

Afterthoughts

What does the function ^{y}x look like for real y, intuitively? Roughly
speaking, it is a growing tower of x's, with y controlling the tower's acquisition of
further exponentials. The highest exponent q = y-[y], always varies in [0,1). The
interesting stuff happens as y approaches a natural number n from the left. When y
approaches n thus, the tower still has [y] + 1 = n - 1 + 1 = n exponentials. I.e. it is
x^{x...xq}, n x's (q = 1^{-}).

Now when y passes over n becoming an integer and then growing further, the last
exponential x^{q} "stabilizes" to x^{1} and a new exponential starts on top of it, with a new q very close to 0. I.e.
x^{x...xq}, n + 1 x's (q = 0^{+}).

The exp function (which is what's working under the scenes) makes the transition seamless, resulting in a smooth transitioning into the exponential tower that has one more exponential at its top.

Intuitively, one could perhaps visualize this function as an already infinite
exponential, in which successive *state exponents* get activated continuously.
I.e.,

(x^{y1})^{(xy2)...(xyn)...}

The state exponents y_{i} of the function are determined uniquely by the
decomposition of y as,

y = [y] + {y-[y]} = n + q, n in N, q in [0, 1), with y_{i} = 1 for i <=
n, y_{n+1} = q, y_{j} = 0, j > n.

y_{i} always ranges in [0,1) and the function starts with y_{i} = 0,
for all i > 1. As y in ^{y}x moves in (0, +∞), it causes (by virtue of
its own unique decomposition above) a continuous activation of those exponents
y_{i}, according to the scheme above, causing successive acquisitions of
further exponentials, which force the tower to grow indefinitely, preserving however
continuously the functions ^{n}x as y passes through the naturals. And that's
what we wanted.

What is also interesting, is that the above continuous extension satisfies the
*hyperexponential-logarithm* property: log(^{y}x) = ^{y-1}x*log(x)
for all y in R, y > 0.

That the property log(^{n}x) = ^{n-1}x*log(x) holds for y in N, can
be shown easily via induction.

Now let's see if the above identity is true for ^{y}x with x, y > 0.
Assume than y is in R^{+}-N. log(^{y}x) = log(F(x, x^{y-[y]},
[y])) (1)

Let [y] = n, y - [y] = q, n in N, q in [0,1).

Then (1) is equal to:

log(x^{x...xq}) (n+1 x's) =

x^{x...xq}*log(x) (n x's) =

F(x, x^{q}, n-1)*log(x) (2)

Now if y is in R^{+}-N, it is true that if [y] = n then [y-1] = n-1,
therefore,

y-1-[y-1] = y-1-(n-1) = y-1-n+1 = y-n = y-[y] = q, and therefore,

^{y-1}x*log(x) =

F(x, x^{y-1-[y-1]}, [y-1])*log(x) =

F(x, x^{q}, n-1)*log(x). (3)

The result follows from (2) and (3) (a more formal proof can be given using induction, which the author omits).

Note that ^{y}x as defined above works for y >= 0 (naturally), but there
is no problem if x is complex. In other words, ^{y}w, y >=0, w in C, works
as expected. In fact, tracing the behavior of ^{y}w can reveal a plethora of
information about the convergence of various iterated exponentials. In particular,
Macintyre in the references argues as follows:

"If we note that the function w=exp(i*π*z/2) maps the half strip 0<Re{z}<1,
Im{z}>0 on to the quadrant |w|<1, 0<arg(w)<π/2, the convergence (of
i^{ii...}) is proved by considering iterations of this
mapping..."

With Maple, the above can be visualized as follows:

>w:=z->exp(I*Pi*z/2);

> for k from 1 to 5 do

> p[k]:=plot([seq([Re((w@@k)(1/10*n)),Im((w@@k)(1/10*n))],n=0..10)]):

> od:

> plist:={seq(p[k],k=1..5)}:

> display(plist);

The Macintyre reference contains a hand-drawn representation of the basin exactly as shown above.

Continuously tracing ^{y}i, for y >=0, produces a surprisingly similar
result:

>complexplot('f_Q(I,y)',y=0..15);

The behavior of f_Q can become quite complex, literally. It is obvious that
^{n}(-1) = -1, for all n in N, however, continuously tracing f_Q(-1,y) can lead
to an infinity of new surprises:

>complexplot('f_Q(-1,y)',y=0..1);

> complexplot('f_Q(-1,y)',y=1..2);

> complexplot('f_Q(-1,y)',y=2..3);

> complexplot('f_Q(-1,y)',y=3..4);

> complexplot('f_Q(-1,y)',y=4..4.05);

Note that while f_Q always returns to -1 (infinitely often, since ^{n}(-1) =
-1, for all n in N), the in between trace is very complex.

The only problem function ^{y}x has, is that it is not made to agree with
the hyperroot/tetraroot function as it was defined in a previous article. However, this is a minor nuisance, because it is rather
the definition of the hyperroot which may be considered deviant in this case.

- For an in-depth examination of the problem consult the Tetration References.
- There is a way to extend tetration to real numbers based on tetraroots which preserves commutativity of the hyper exponents when both exponents are rationals and reciprocals of each other. For details, consult this article.
- For the actual details of extending the fundamental exponentials,
G
_{a}(x) = a^{x}and H_{a}(x) = x^{a}continuously over the reals, the reader should consult "The Number Systems, Foundations of Algebra and Analysis", by Solomon Feferman, page 285. - Compare with the graphs in the Tetration References.