Content uploaded by Yiannis Galidakis

Author content

All content in this area was uploaded by Yiannis Galidakis on Sep 23, 2020

Content may be subject to copyright.

Content uploaded by Yiannis Galidakis

Author content

All content in this area was uploaded by Yiannis Galidakis on Sep 11, 2020

Content may be subject to copyright.

The Problem of the Inﬁnite Exponential

Ioannis N. Galidakis,

Mathematician,

Agricultural University,

Athens, Greece

August 2020

Dissertation for the title of “Doctor of Mathematics” at the Agricultural University

of Athens, submitted by Ioannis Galidakis, Mathematician from Athens, in August 30,

2020.

1

Advisor: Professor Ioannis Papadoperakis

Co-advisors: Professors Dimitrios Gatzouras, Charalambos Charitos

2

To the memory of my Parents

3

Contents

1 Introduction 5

2 An application of function Wto inﬁnite exponentials 6

2.1 Deﬁnitions.................................... 6

2.2 TheWfunction................................. 7

2.3 Thecentrallemma ............................... 10

2.4 Convergence for x∈R............................. 11

2.5 Convergence for c∈C............................. 14

2.6 Convergence for q∈Q............................. 15

3 An application of functions HW to inﬁnite exponentials 16

3.1 Deﬁnitions.................................... 16

3.2 The central lemma on HW . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Topology of the inﬁnite exponential 18

4.1 JuliaandFatousets .............................. 18

4.2 Attractors and repellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 General topological map of the inﬁnite exponential . . . . . . . . . . . . . 20

4.4 Topology .................................... 22

4.5 Fractal nature in the inﬁnite exponential . . . . . . . . . . . . . . . . . . . 22

4.6 The primitive of a Cantor bouquet . . . . . . . . . . . . . . . . . . . . . . 26

4.7 Pinball with the inﬁnite exponential . . . . . . . . . . . . . . . . . . . . . 27

5 The generalized inﬁnite exponential 27

5.1 Deﬁnitions.................................... 27

5.2 Ascendingindexes ............................... 28

5.3 Descendingindexes............................... 29

6 Appendix: programming/results 31

7 Acknowledgements 33

8 Bibliography 34

4

1 Introduction

The problem of the inﬁnite exponential was observed and studied for the ﬁrst time by

Leonhard Euler ([85]). It concerns the sequence deﬁned recursively as follows:

Deﬁnition 1.1 Given x > 0,

αn+1 =xαn

The central problem of the inﬁnite exponential is, then, the Proposition:

Proposition 1.2 Given x > 0, study the behavior of the recursive sequence αn,n∈N.

Euler proved the following theorem:

Theorem 1.3 The sequence αnconverges if and only if x∈e−e, ee−1.

The central problem leads to an extensive labyrinth of sub-problems, which are ex-

amined partially or fully in the bibliography references. The reason why the area of

sub-problems is deep and extensive, is that the sequence (1.1) is based on the arith-

metic operator of exponentiation, which is the most “powerful” of the three {+,×,↑},

but unfortunately it is the only one that is not commutative. In other words, in general1,

ab̸=ba.

A second reason the central problem leads to a plethora of sub-problems, is that

the two previous operators do not exhibit any idiopathic behavior relative to their re-

peated application. For example, for given x > 1, the repeated operations +(n)x=

x+x+...+x

n

and ×(n)x=x×x×. . . ×x

n

, cause divergence. The latter does not al-

ways happen with the exponentiation operator, as the sequence (1.1), may converge as

with Euler’s Theorem, for x∈e−e, ee−1.

The above diﬀerence becomes immediately obvious for example, when a student at-

tempts to investigate the behavior of the sequence (1.1), for given x > 1, let’s say for

x= 1.2, where a simple pocket calculator shows that the repeated exponentiation of such

xleads to an unexpected convergence.

This unexpected convergence is the exceptionally strange of theorem (1.3), something

which was observed by Euler, using only paper and pencil. Something like that is really

inconceivable without the use of computers, at least insofar modern calculation meth-

ods are concerned, as the exponentiation operator exhibits mainly calculation diﬃculties

which do not exist when using the other two previous operators, since repeated addition

and multiplication are relatively “fast” operations when using computers.

We, then have a problem which could possibly be investigated further using fairly

strong computer systems, which could conceivably help with many of the sub-problems

that show up.

The ﬁrst important development came around 1999-2000, when fairly advanced ver-

sions of the Computer Algebra Systems (CAS) Maple and Mathematica showed up [154].

1The problem of solving the equation ab̸=ba, which is thus of substantial importance in the inves-

tigation of the inﬁnite exponential, was presented for the ﬁrst time to the author in 1982, in a letter of

the author’s Father to the author

5

These packages include since then symbolic versions of the complex function W, otherwise

known as Lambert’s function. This latter function has found multiple applications the

last ten to ﬁfteen years in various sectors of pure and applied mathematics, like physics,

electronics, optics and astronomy.

The function W is a limited inverse of the complex function z·ez, with the latter

being intrinsically connected to the operator of the repeated exponentiation, because the

last one can be analyzed in parts by z·ez, when one considers the partial sub-exponents

of a repeated exponential, using the primitive base exp.

Finally and after years of research in arithmetical analysis of the results of the survey

related to the inﬁnite exponential and the Maple CAS Maple ([67, 332]), it is validated

that the function W, gives a closed form for the convergence limit of the sequence (1.1)

as it follows from Theorem (1.3), when the sequence (1.1) converges.

Therefore the function W becomes the ideal tool to examine the inﬁnite exponential

with and consequently allows a direct extraction of ﬁnal conclusions related to the area

of convergence of the sequence (1.1), extending Theorem (1.3) to the complex plane, to

an area called today area Shell-Thron.

The above results are generalized directly to the space of Quaternions, with the new

results coinciding with the already known ones for the complex plane, despite the Quater-

nion space not being (in general) commutative.

The problem of the inﬁnite exponential does not end there, as arithmetic calculations

also suggest non-convergence or even divergence of the sequence in many sub-areas of the

complex plane, which need additional calculations in order to ﬁnd out if in these areas

the sequence (1.1) behaves normally and how.

Examining these areas of unknown behavior of the inﬁnite exponential, requires the

use of functions which are gradually more complex in nature than W and the author calls

them HW. These functions solve additionally the cases where the sequence (1.1) falls

into cycles of period p > 1, characterizing therefore fully the sub-areas of convergence of

the sequence, when there are multiple attractors, which are given in closed form by the

functions HW, and determine therefore the partial limits of the convergence.

This work collects sub-results until Proposition (1.2) of Euler is answered in its general

form, which is essentially the topology of the sequence αn+1 =γαn

n, for general γn∈C,

calculates a dimension bound for its main feature set and examines partially the behavior

of the functions HW which allow the extraction of conclusions for this Proposition.

2 An application of function Wto inﬁnite exponentials

2.1 Deﬁnitions

The function W acquired substantial publicity lately, mainly because of important progress

in computational mathematics. Even though compositions of it appear in hidden form

in references [18, 153], [187, 14] and [122, 235], its essential properties are presented in

[67, 344-349] and [68, 2-8]. Some of these properties can be used to simplify the answer

of when the inﬁnite exponential converges.

We are working with the principal branch of the complex map log, and we use Maurer’s

notation for repeated and inﬁnite exponential (see [122, 239-240]).

6

It is assumed that equations with complex exponents anywhere in this dissertation

use always the principal branch of the complex exponentiation, whenever this is needed:

cw=ew·log(c),c̸= 0, with log always denoting the principal branch of the complex

logarithm log(k, z), for θ∈(−π, π].

Deﬁnition 2.1 For z∈C\ {x∈R:x≤0}and n∈N,

nz=z, if n= 1,

z(n−1z), if n > 1.

Whenever the following limit exists and is ﬁnite, we set:

∞z= lim

n→∞

nz(1)

We use the exponential map gc(z) and its compositions, for c∈C\ {x∈R:x≤0}.

Whenever the parameter cis obvious, it will be omitted to avoid any confusion.

gc(z) = cz(2)

g(n)

c(z) = gc(z) if n= 1,

gcg(n−1)

c(z)if n > 1.(3)

nzand g(n)(z) are related: nc=g(n)

c(1). We also use the following function, which is

a partial inverse of W:

m(z) = z·ez, z ∈C(4)

The terminology Inﬁnite Exponential seems to have been used ﬁrst in [18, 150]. In

short, it is the inﬁnite tower zzz···

3

2

1, with zn∈R(or zn∈C), ∀n∈N. We are initially in-

terested in the case zn=z,∀n∈N. Respectively, the terminology zz···z

(real or complex)

will be used alternatively to to the notation z(z(···z))to notate repeated exponentiation

from the top to the bottom. In the majority of the cases we use Deﬁnition (2.1) for the

repeated exponential.

2.2 The Wfunction

The complex function W is a partial inverse of m(z) or otherwise the function which

solves the equation m(z) = wrelative to z. Alternatively,

Deﬁnition 2.2 Wsatisﬁes the functional equation:

W(z)eW(z)=z, z ∈C

7

Figure 1: Quadratrix of Hippias bounds the ranges of W(k, z), k∈Z

W is multivalued and has inﬁnitely many branches as a complex Riemann surface. It

is usually notated as W(k, z), with k∈Zspecifying the working branch. Speciﬁcally, its

principal branch W(0, z) corresponds to k= 0, and then the function is notated as W(z).

Follow some useful properties of W. Most of these can be found in [67] and [68] and

can be validated numerically using the Maple CAS in [154, 305]. We selectively prove

only some of the properties which are not immediately obvious in [68] or [67].

For k∈Z, the various branches of W(k, z ) are deﬁned in Cand are discontinuous at

the points of the intervals BCk:

BCk=

−∞,−e−1, if k= 0

−∞,−e−1∪−e−1,0, if k=−1

(−∞,0) , otherwise

We observe that the branch point z0=−e−1of W(z) is m(z), with zsatisfying:

dm

dz = 0. This branch point is shared between the two branches W(z) and W(−1, z).

Suppose then that CN = (−∞,−1) and let us consider the curves:

Ck=−ycot(y) + yi, y ∈(2kπ, (2k+ 1)π), if k≥0,

−ycot(y) + yi, y ∈((2k+ 1)π, (2k+ 2)π), if k < 0.

Lemma 2.3 The image of BCkunder W(k, z )is:

W(k, B Ck) = C−1∪CN, if k=−1

Ck, otherwise

We now deﬁne the regions Dkas follows:

Dk=

area between C1, CN , C0, if k= 1

area between C−1, CN , C−2, if k=−1

area between Ck, Ck−1, otherwise

8

The bounding curves of the ranges of W(k, z)Ckand the regions Dkgive the well

known quadratrix of Hippias, which is presented in [68] and in Figure 1.

The range of the images of W is then constrained by the regions Dk(see [67, 13-23]),

consequently,

Lemma 2.4 W(k , C\ {0}) = Dk∪Ck,k∈Z\ {0}, and W(0,C) = D0∪C0

W(k, z) is a partial inverse of m(z) in the aforementioned regions Dk∪Ck, consequently

(see [67])

Lemma 2.5 W(k , m(z)) = z,k∈Z,z∈Dkand m(W(k, z)) = z,k∈Z,z∈C

Corollary 2.6 W(m(z)) = z,z∈D0and m(W(z)) = z,z∈C

From the symmetry between Ckand Dkfollows:

Lemma 2.7 W(k , z) = W(−k, z),k∈Z,z∈C

Corollary 2.8 W(z) = W(z),z∈C

Since (−∞,−1] ∈D−1and [−1,+∞)∈D0, only the branches that correspond to

k= 0 and k=−1 can ever assume real values.

Lemma 2.9 W(k , z)∈R⇒k∈ {−1,0}

From the ﬁrst diagram in [67] (or elementary calculus), follow,

Lemma 2.10 W(x)is real, continuous and monotone increasing on the interval [−e−1,+∞).

Lemma 2.11 W(−1, x)is real, continuous and monotone decreasing on the interval

[−e−1,0).

Lemmas (2.10) and (2.11) follow easily by considering the function m(z) and the fact

that −e−1is a common branch point between W(z) and W(−1, z):

Lemma 2.12 W(e) = 1

Lemma 2.13 W−e−1= W −1,−e−1=−1

Lemma 2.14 D⊂D0and ∂D ∩∂D0={−1}

Proof: It suﬃces to show |−ycot(y) + yi|>1 for each y∈0,π

2. We observe

immediately that lim

y→0+(−ycot(y) + yi) = −1∈∂D ∩∂D0and z∈∂D ∩∂D0⇒

cos(π−y) + sin(π−y)i=−ycot(y) + yi ⇒ {sin(y) = y, −ycot(y) = cos(π−y)}. From

the ﬁrst equation we get y=kπ,k∈Z. From it only the equation y= (2k+ 1)π,k∈Z

satisﬁes the second equation as a limit, therefore z= cos((2k+1)π)+ sin((2k+1)π)i=−1

and the Lemma follows.

9

Lemma 2.15 W(z)is analytic at the origin z0= 0 with series:

S(z) = ∞

n=1

(−n)n−1zn

n!

and convergence radius: Rs=e−1

Proof: Details related to the series S(z) as well as other expansions are given in [67].

The Ratio test reveals the radius of convergence. lim

n→∞

an+1

an= lim

n→∞ 1 + 1

nn−1z=

|ez|<1, or equivalently, |z|< e−1.S(z) is valid on the entire disk Dw=z:|z| ≤ e−1.

When |z|=e−1, then

(−n)n−1zn

n!=nn−1

enn!<nn−1

√2πn(n+1

2)=1

√2πn 3

2, using Stirling’s ap-

proximation and the series ∞

n=1

1

√2πn 3

2converges.

The fact that S(z) has radius of convergence Rs=e−1, follows also from the fact that

W(z) has a branch point at z0=−e−1.

2.3 The central lemma

the main result of Lemma (2.18) which concerns the limit of the Euler sequence is men-

tioned in [67, 332]. The author thinks it deserves a deeper analysis, particularly after the

fact that W is a multivalued function.

g(z) of (2) is intricately connected with the inﬁnite exponential. Generally speaking,

given c∈C,c /∈ {0,1}, if the sequence g(n)(z), n∈Nconverges, then it must converge

to a ﬁxed point of g(z) or equivalently the limit must satisfy the ﬁrst auxiliary equation,

z=g(z) (5)

Equation (5) can always be solved through W.

Lemma 2.16 The ﬁxed points of g(z)are given by h:Z×C7→ C, with:

h(k, c) = W(k, −log(c))

−log(c),k∈Z

Proof: z=g(z)⇔z=cz⇔ze−zlog(c)= 1 ⇔ −zlog(c)e−zlog(c)=−log(c)⇔

m(−zlog(c)) = −log(c)⇔ −zlog(c) = W(k, −log(c)), k∈Z, from the Deﬁnition of

m(z), ⇔z=W(k,−log(c))

−log(c),k∈Zand the Lemma follows.

Lemma 2.17 If c∈C\ee−1, then h(k, c)is a repeller of g(z).

Proof: If c̸=ee−1, then g′(h(k , c)) = log(c)W(k,−log(c))

−log(c)=−W(k, −log(c)) ∈ −Dk,

from Lemma (2.4), therefore if k̸= 0, then |g′(h(k , c))|>1, from Lemma (2.12), and the

Lemma follows.

If c=ee−1, then g′(h(−1, c) = −W(−1,−e−1) = −1 (Lemma (2.11)).

The supposition c̸=ee−1is important. Otherwise, g′(h(k, c)) = −Wk, −e−1= 1,

for k=−1 (Lemma (2.17)).

Lemmas (2.16) and (2.17)lead to the central Lemma of the dissertation:

10

Lemma 2.18 (Corless) Whenever the limit of g(n)(z)exists ﬁnitely, its value is given

by:

h(c) = h(0, c) = W(−log(c))

−log(c)

therefore

lim

n→∞ g(n)(z) = lim

n→∞ g(n)(1) = ∞c=h(c)

2.4 Convergence for x∈R

The fact that the repeated iterated exponential with base √2 converges to 2 can be

validated arithmetically in [70, 70] and [117, 66] and is explained analytically in [171,

434], [127, 77] and [139, 643-646]. Considering the following relation for the positive

square root of 2, then (√2)2= 2. replacing the exponent of the relation with the left side

of the equation, we get the sequence g(n)(2), n∈N. From the previous Lemma when the

limit exists, limn→∞ g(n)(z) = 2, n∈Nfor suitable z, consequently,

∞(√2) = 2 (6)

This will also be shown with the last Lemma of this section.

Lemma 2.19 If x∈e−1, e, then hxx−1=x.

Proof: By deﬁnition, h(y) solves the equation x=yxwhich is equivalent to xx−1=y,

therefore it is a partial inverse of y(x) = xx−1. In the given interval y(x) is 1-1 and onto

the range e−1, e, and the Lemma follows.

Lemma 2.20 If x∈(e, +∞), then hxx−1=w∈(1, e), with ww−1=xx−1.

Proof: y(x) is continuous on (1,+∞), acquires a maximum at x=e, and lim

x→∞ y(x) = 1,

therefore there exists unique win (1, e), such that ww−1=xx−1, and the Lemma follows

from lemma (2.19).

Example: y= 1.3304 .

= 1.562(1

1.562 ).

= 6.620(1

6.620 ). Such values are given in closed form

using W, but can also be given arithmetically or using other methods in the references

which deal with the solution of the equation xy=yx, like [43, 763], [55, 222-226], [56,

78-83], [89, 137], [141, 233-237], [161, 316], [169, 444-447], [183, 141], and [125, 96-99]. In

the references the authors observe that y(x) is a partial inverse of h(x), but don’t deﬁne

hthrough W.

Lemmas (2.19) and (2.20) in short:

Lemma 2.21

hxx−1=x, if x∈e−1, e;

w, w ∈(1, e) : ww−1=xx−1,if x∈(e, +∞).

11

Lemma 2.22 If c=e−e,x0= log Wlog(c)−1log(c)−1log(c)−1, and u(x) = g(2) (x)−

x, then,

x0is the only critical point of u(x), in [0,1] (7a)

x0=e−1(7b)

u(x0) = 0 (7c)

du

dxx0

= 0 (7d)

du

dx <0,x∈[0,1] − {x0}(7e)

Proof: du

dx = 0 can be solved exactly through W. If du

dx = 0, then g(2) (x)g(x) log(c)2= 1,

therefore eylog(c)ylog(c)= log(c)−1, where y=cx. Then m(ylog(c)) = log(c)−1, therefore

ylog(c)= W k, log(c)−1, consequently y= W k , log(c)−1log(c)−1, and ﬁnally x=

log Wk, log(c)−1log(c)−1log(c)−1,k∈Z. (7a) follows directly from Lemma (2.5).

(7b) follows from Lemma (2.11). (7c) and (7d) follow easily. For (7e) we observe that

log(c) = −e < 0, therefore, if x < x0then g(x)> e−1and g(2)(x)< e−1, therefore

g(2)(x)g(x) log(c)2<1, consequently du

dx <0. For x > x0the proof (with inequality signs

reversed) is similar and the Lemma follows.

Lemma 2.23 If c∈e−e, ee−1, then ∞c=h(c).

Proof: If c=e−e, the ﬁxed point of g(x) is given by Lemma (2.18). h(c) = h(e−e) =

W(−log(e−e))

−log(e−e)=W(e)

e=e−1, based on Lemma (2.26). Using Lemma (2.22), the continuity

of u(x) and the factoids: u(0) = c > 0, u(1) = cc−1<0, it follows that g(2) (x)> x, if

x∈0, e−1and g(2)(x)< x, if x∈e−1,1. Using the two inequalities and induction

on n, the sequence: an=g(n)(1), n∈Nsatisﬁes, a2n+2 < a2n, and a2n+3 > a2n+1, for

each n∈N. The last shows that a2n+1 and a2nare monotone increasing and decreasing,

respectively. Additionally, since 0 < c =e−e<1, both sequences are bounded above by

1 and below by 0. It follows that both sequences a2n+1 and a2npossess limits. Since the

only root of u(x) is x0(otherwise equation (7a) is violated), both sequences converge to

x0, from which follows that anconverges to x0=e−1.

If c=ee−1, then the ﬁxed point of g(x) is given again by Lemma (2.18). h(c) =

hee−1=W(−log(ee−1))

−log(ee−1)=W(−e−1)

−e−1=e, based on Lemma (2.27). Using induction on

n, the sequence an=g(n)(1), n∈Nis monotone increasing and bounded above by e,

therefore it converges to eand the Lemma follows.

[122, 240], [18, 153] and [187, 14-15] arrive at the same result diﬀerently, without using

the function W.

Lemma 2.24 If c∈(0, e−e), then ∞cdoes not exist.

Proof: The ﬁxed point h(c) of g(x) from Lemma (2.18) is a repeller. If c∈(0, e−e), then

W(e)<W(−log(c)) based on Lemma (2.10) and consequently 1 <W(−log(c)), based

on Lemma (2.26). This means 1 <|g′(h(c))|and the Lemma follows from ﬁxed point

iteration.

12

Lemma 2.25 If c∈e−e, ee−1, then ∞c=h(c).

Proof: The ﬁxed point h(c) of g(x) from Lemma (2.18) is an attractor. If c∈e−e, ee−1,

then −e−1<−log(c)< e, therefore W −e−1<W(−log(c)) <W(e), based on Lemma

(2.7) and consequently |g′(h(c))|<1 based on Lemmas (2.10) and (2.11) and the Lemma

follows from ﬁxed point iteration.

Lemma 2.26 If c∈ee−1,+∞, then ∞cdoes not exist.

Proof: The ﬁxed point h(c) of g(x) from Lemma (2.18) is a repeller. If c∈ee−1,+∞,

then −log(c)∈BC0, therefore W(−log(c)) ∈C0, based on Lemma (2.12), and conse-

quently |g′(h(c))|>1, based on the same Lemma and the Lemma follows from ﬁxed

point iteration.

Lemmas (2.23)-(2.26) and (2.18) validate the ﬁnal Lemma of this section which is

Theorem (1.3) of Euler:

Lemma 2.27 ∞cexists, then and only, when c∈e−e, ee−1, me ∞c=h(c).

Proof: The interval of convergence can be determined for the real case from ﬁxed

point iteration. The only potential ﬁxed point of g(x) is given from Lemma (2.18), as

h(c). Using elementary properties of all the relevant functions, if |g′(h(c))| ≤ 1, then

|−W(−log(c))| ≤ 1. This means W(−log(c)) ∈[−1,1], therefore m(W(−log(c))) ∈

m([−1,1]), or m(W(−log(c))) ∈−e−1, e, therefore also −log(c)∈−e−1, e, based on

Deﬁnition (2.2) and ﬁnally c∈e−e, ee−1.

Using Lemma (2.27) for c=ee−1,c=e−eand c=√2,

∞ee−1=ee−1(ee−1)···

=hee−1=e

∞e−e=e−e(e−e)···

=he−e=e−1

∞√2=√2(√2)···

=h21

2= 2

The last equations settle normally the question in the beginning of this section with

equation (6). That the inﬁnite exponential converges if and only iﬀ its base belongs to

the interval e−e, ee−1.

= [0.06598,1.44466] is also shown in [122, 240], [139, 645] and

[146, 556] using other methods and without employing the function W.

[9, 207-208] and [135, 301-303] also show that for k∈N, lim

c→0+

2kc= 1 and lim

c→0+

2k+1c=

0. If c∈(0, e−e), then nc,n∈Nis a cycle of period 2 by considering the even and odd

subsequences. The bifurcation that occurs and its behavior are analyzed in [9, 207], [187,

15] and [135, 299]. We note that the two branches which stem from the bifurcation point

e−e, e−1can be given in parametric form as a(a

1−a)and a(1

1−a)for proper positive a

(see for example [122, 237] or [188, 212]). In this case, as is shown in [171, 434], [122,

13

241-243], [187, 13] and [129, 501], the two separate limits a= lim

n→∞

2n+1cand b= lim

n→∞

2nc

satisfy the inequality 0 < a < h(c)< b < 1 and the second auxiliary system,

a=cca

b=ca(8)

A closed form solution for system (8) is given in detail in a subsequent section, where

the diﬃculties of solving the auxiliary equation of order nare presented, for the complex

exponential map g(z), using certain generalized functions based on W.

2.5 Convergence for c∈C

Suppose Dis the unit disk. We consider the map ϕ:C7→ C, deﬁned as: ϕ(z) = e(z

ez)=

e−m(−z). The image RS T =ϕ(D) is a nephroid region called Shell-Thron region, with

an approximation given in [210] and in Figure 2.

Figure 2: Shell-Thron region

[165, 679], [164, 12] and [13, 106] show that on the interior of RST we have convergence

of g(n)(z). What happens at the boundary of the ﬁgure is mentioned in [12, 502] and [11]:

Theorem 2.28 (Baker/Rippon) The sequence nc,n∈Nconverges for log(c)∈ {te−t:

|t|<1, or tn= 1, for some n∈N}and diverges elsewhere.

tand care related through W, by considering always the principal branches of all

maps involved. c=ϕ(t)⇔W (−log(c)) = W (m(−t)) ⇔t=−W (−log(c)), using

Corollary(2.6) and Lemma (2.12). Consequently ϕis reversible and ϕ−1= (−W)◦(−log)

with t=ϕ−1(c). Then we have the Theorem:

Theorem 2.29 The sequence nc,n∈Nconverges if ϕ−1(c)<1or ϕ−1(c)n= 1,

where ϕ−1=−W(−log(c)), some n∈Nand converges elsewhere.

Lemma 2.30 If c∈C, then the multipler of the ﬁxed point h(c)of g(z)is given by

t=ϕ−1(c).

14

Proof:

g′(h(c)) = log(c)·g(h(c))

= log(c)·h(c)

= log(c)·W (−log(c))

−log(c)

=−W(−log(c)) = t

and the Lemma follows.

Theorem (2.28) then alternatively states that when c=ϕ(t), then the sequence g(n)(c),

n∈Nconverges if and only if the measure of the multiplier t=ϕ−1(c) of the ﬁxed point

h(c) of g(z) is less than 1, or the multiplier is a n-th root of unity.

Lemma 2.31 If |t|<1and c=ϕ(t), then ∞c=h(c).

Proof: The ﬁxed point of g(z) is given by Lemma (2.18), as h(c). h(c) = W(−log(c))

−log(c)=

W(−te−t)

−te−t=W(m(−t))

−te−t=−t

−te−t=et, using Lemma (2.4) and Corollary (2.6). The ﬁxed

point h(c) = etof g(z) is an attractor. According to Lemma (2.30), |g′(h(c))|=|t|<1,

which is true by assumption and the Lemma follows from ﬁxed point iteration theory and

Lemma (2.18).

Lemma 2.32 If |t|>1and c=ϕ(t), then ∞cdoes not exist.

Proof: The ﬁxed point h(c) = etof g(z) is a repeller: According to (2.30), |g′(h(c))|=

|t|>1, which is true by assumption and the Lemma follows similarly.

More applications for this section, like Fractals based on repeated exponential iteration

and more identities, may be veriﬁed in [210].

2.6 Convergence for q∈Q

Generalizing the function W = Wcas Whin the space Q, we note that convergence still

happens. First we get a generalization of the Theorem of Shell ([164]).

Lemma 2.33 If q∈Q, then the sequence nq,n∈Nconverges if t=ϕ−1(q)∈(S3)◦.

Lemma 2.34 If q∈Qand the sequence g(n)(q)converges, the value of the limit is always

given as:

h(q) = Wh(−ln(q))

−ln(q)

In the space of Qthe function ln is not (in general) commutative, therefore we have

two convergence orbits, however the orbits ﬁnally converge to the same point given by

Lemma (2.34) which serves as a generalization of Lemma (2.18).

Examples: If q= 1/2−1/2i+1/2j−1/2kthen |tq|=|−WH(−log(q))|.

= 0.70724 <1,

therefore tq∈(S3)◦, therefore according to Lemma (2.33) the sequence nqconverges. The

15

ﬁrst 14 values with Maple are given as:

1q.

= 0.5−0.5i+ 0.5j−0.5k

2q.

= 0.34967 −0.11655i+ 0.11655j−0.11655k

3q.

= 0.75577 −0.16732i+ 0.16732j−0.16732k

4q.

= 0.51884 −0.30319i+ 0.30319j−0.30319k

5q.

= 0.49389 −0.17222i+ 0.17222j−0.17222k

6q.

= 0.63600 −0.20888i+ 0.20888j−0.20888k

7q.

= 0.53831 −0.24422i+ 0.24422j−0.24422k

8q.

= 0.54276 −0.19809i+ 0.19809j−0.19809k

9q.

= 0.58839 −0.21696i+ 0.21696j−0.21696k

10q.

= 0.55060 −0.22511i+ 0.22511j−0.22511k

11q.

= 0.55731 −0.20924i+ 0.20924j−0.20924k

12q.

= 0.57094 −0.21767i+ 0.21767j−0.21767k

13q.

= 0.55692 −0.21898i+ 0.21898j−0.21898k

14q.

= 0.56109 −0.21373i+ 0.21373j−0.21373k

The central Lemma (2.34) gives the limit: ∞q=h(q).

= 0.56182−0.21640i+0.21640j−

0.21640k.

If q= 1 + i+j−kthen |tq|=| −WH(−log(c))|.

= 0.94126 <1, therefore tq∈(S3)◦,

consequently again based on Lemma (2.33) the sequence nqconverges also. Lemma (2.34)

gives the limit: ∞q=h(q).

= 0.46827 + 0.33791i+ 0.33791j−0.33791k.

Details and extensive analysis for the results in the space Qare given in [207].

3 An application of functions HW to inﬁnite exponen-

tials

Divergence in the previous sections doesn’t necessarily mean that |∞c|=∞. Divergence

may be such that the limit is ∞, but can also be periodic and even totally chaotic. The

periodic points given by (2.18) all have period 1, so it follows that these are automatically

solutions to the n-th auxiliary equation g(n)(z) = z, but not all solutions are given only

from the main Lemma. Such points exist. To determine such points we need functions

which are stronger than W.

3.1 Deﬁnitions

Suppose fi(z) are non-vanishing identically complex functions. We deﬁne Fn,m(z) : N2×

C→Cas:

Deﬁnition 3.1

Fn,m(z) = ez, if n= 1,

ecm−(n−1)Fn−1,m (z), if n > 1.

16

Deﬁnition 3.2 G(f1, f2, . . . , fk;z) = z·Fk+1,k+1(z)

If k= 0, then G(z) = z·ez. If k= 1 then G(c1;z) = zec1ez. If k= 2 then G(c1, c2;z) =

zec1ec2ez

. When we write about the HW, we can use the terminology G(. . . ;z), meaning

that the corresponding function includes meaningful terms-parameters. The function of

interest here is the inverse of G(. . . ;z).

HW(f1, f2, . . . , fk;y) (9)

In other words Gand HW satisfy the functional relation:

G(. . . ; HW(. . . ;y)) = y(10)

by supposing always that the list of parameters is identical on both sides. These maps

will henceforth be called HW functions. We note that when k= 0, HW(y) = W(y) is the

function W. The existence of the HW is guaranteed in all cases by the Lagrange Inversion

Theorem (see [159, 201-202]).

3.2 The central lemma on HW

A generalization is now immediate, so we are close to a variant of the central Lemma.

Lemma 3.3 If c∈C\ {0,1}, then the p-th auxiliary equation f(p)(z) = z, admits the

solutions z, f (z), . . . , f(p−1)(z), where

z=HW(−log(c), . . . , log(c); log(c))

log(c)(pparameters)

Proof: Similarly to the central Lemma, the functions HW solve the p-th equation

g(p)(z) = zas follows:

g(p)(z) = z⇒

cc...cz

=z⇒

z·c−c...cz

= 1 ⇒

z·e−log(c)e...elog(c)z

= 1 ⇒

zlog(c)·e−log(c)e...elog(c)z

= log(c)⇒

zlog(c) = HW(−log(c), . . . , log(c); log(c)) ⇒

z=HW(−log(c), . . . , log(c); log(c))

log(c)p parameters

We note that if zis a solution of g(p)(z) = z, then if k∈ {1,2, . . . , p −1}we also have

g(p)(g(k)(z)) = g(k)(g(p)(z)) = g(k)(z) and the Lemma follows.

Corollary 3.4 Whenever nc,n∈Nis an attracting cycle, the limits of the psubsequences

n+kc,n∈N,k∈ {0,1, . . . , p −1}are given as z, g(z), . . . , g(p−1)(z), with zbeing given

by the expression of Lemma (3.3).

17

Finally,

Lemma 3.5 If c∈C\RS T and t= HW(−log(c), . . . , log(c); log(c)) (pparameters),

then the sequence nc,n∈Nis an attracting cycle of period p, if tetp−2

k=1 f(k)(et)<

log(c)−p+1and only if tetp−2

k=1 f(k)(et)≤log(c)−p+1.

Proof: The multiplier of the ﬁxed point zof the function g(n)(z) is given in [209], as:

g(n)′(z) = (log(c))n·

n

k=1

g(k)(z) (11)

The ﬁxed point zis given by the expression of Lemma (3.3), through the HW function,

so substituting this point in equation (11), the multiplier takes the form:

(log(c))p−1·tet

p−2

k=1

g(k)(et) (12)

The attractor condition of the function g(p)(z) at the ﬁxed point zis g(n)(z)′≤1,

therefore the Lemma follows from equation (12).

Example: Can we say anything for example about the sequence nc,n∈N, if c=−1+i?

We can use the code of the Appendix to check the measure of the corresponding multipliers

tpfor p-cycles. t1=ϕ−1(c) = −W(−log(c)) .

= 1.13445, therefore it cannot be a cycle

of period 1 (in other words we don’t have convergence). t2.

= 0.80847 + .33448i, and

|t2et2log(c)|.

= 4.67684, therefore it cannot be a cycle of period 2. t3.

= 0.03281−0.08534i

and |t3et3f(et3) log(c)2|.

=.94227, therefore it has to be necessarily a cycle of period

3. From Corollary (3.4), z= HW(−log(c),log(c); log(c))/log(c).

=−0.03344 −0.01884i,

therefore the three limits of the cycle are given as z, g(z), g (2)(z). This set is calculated

based on the code with accuracy to 5 decimal digits as, −0.03344 −0.01884i, 1.02959 −

0.08808i,−1.29112 + 1.19363i. Convergence to these points is spiral like (see [209]). The

HW maps are studied separately and the exact mechanism which extracts the roots is

shown in [205].

4 Topology of the inﬁnite exponential

4.1 Julia and Fatou sets

To study generally (and speciﬁcally) the behavior of the sequence of Euler, we study

the corresponding topology which results on the complex plane from repeatedly iterating

the exponential map, for diﬀerent (but speciﬁc each time) base of the exponential, since

Euler’s sequence essentially reduces to repeated iteration of some suitable exponential

map, for a speciﬁc base λ= log(c), with c∈Cand initial value z0∈C.

This last topology is the union of two notable sets of the exponential family, that is

the Julia set J[gc] and the Fatou set F[gc] of the exponential, which are complementary

on the complex plane.

18

To characterize therefore the topology of the sequence of Euler, it suﬃces to charac-

terize exactly the sets J[gc] and F[gc] of the exponential map. This characterization has

been done however in the references for the family of maps Eλ(z) = λ·ez, with λ= log(c).

Speciﬁcally, the Julia set of the exponential map Eλ(z) = λ·ezis given in [76], [75]

and [77] and is the complement of the set of periodic repellers of Eλ(z), that is, J[Eλ] =

z∈C:E(n)

λ(z)→ ∞. The Fatou set of the exponential map Eλ, is the complement of

the set J[Eλ] or the set of its periodic attractors, that is F[Eλ] = z∈C:E(n)

λ(z)≤M.

Proposition 4.1 The Julia and Fatou sets of the topology of the inﬁnite exponential,

coincide with the corresponding sets of the repeated iteration of the exponential map

Eλ(z) = λ·ez, with λ= log(c).

Proof: In the aforementioned references it is proved that the Julia set J[Eλ] of the family

of exponential maps Eλ(z) = λ·ezis a Cantor bouquet for 0 < λ < 1/e, therefore it suﬃces

to show that in our case the set J[gc] = J[cz] is in general such a bouquet. Note that the

iteration orbit in the references is E=λ·ez, λ ·eλ·ez, . . .. This orbit can be written

alternatively also as E=λ·ez, λ ·(eλ)ez, . . ., therefore setting λ= log(c), the orbit

is equivalent to the orbit Eλ=log(c)=log(c)·ez,log(c)·cez, . . .. For z= 0, we get the

orbit of zero, as E0={log(c),log(c)·c, . . .}. We observe that eE0=(1,)c, cc, ccc, . . .,

which is the simplest orbit of the Euler sequence for the inﬁnite exponential, therefore

the set J[g] will be exactly what the set JEλ=log(c)is, on the complex plane λ= log(c).

Speciﬁcally, it is a Cantor bouquet. Since the orbits are coincident modulo λ, the same

will hold also for the Fatou sets and the proof is complete.

4.2 Attractors and repellers

We classify now the topology of speciﬁc sub-cases for given λ= log(c) in a neighborhood

Bδ,δ > 0 of the boundary of the Shell-Thron region, by period.

Proposition 4.2 When for λ= log(c)the set J[g]is a Cantor bouquet p-furcation,

then, if there are attractors in the set F[g], they are given from the expression A1=

W(−λ)/(−λ)if p= 1 or from the expression Ap=gk(z0), z0= HW(...,λ)/λk∈

{0,1, . . . , p −1}, if p≥2.

Proof: The proof is done inductively depending on the value of λ= log(c) on the complex

plane, meaning, depending on the position of c=eλrelative to the region RST .

•c∈(RST ∩Bδ(ϕ(t)))◦. Setting λ= log(c) = log(ϕ(t)) = t/et, by assumption is

0<|t|<1, therefore we get the sub-case 0 <|λ|=|t/et|<1/e. In this case, the

sequence of the inﬁnite exponential converges to A1=h(0, c) = −W(−λ)λand the

set J[gc(z)] is a Cantor bouquet. The set F[gc(z)] then is the complement of the set

J[gc(z)] on the complex plane and contains exactly one attractor A1, which drives

any point of F[gc(z)] to A1.

•c∈∂RS T ∩Bδ(ϕ(t)). Setting λ= log(c) = log(ϕ(t)) = t/et, by assumption is

|t|= 1, therefore we get the sub-case |λ|=|t/et|. In this case, when p= 1, then

t= 1 and |λ|= 1/e. The set J[gc(z)] then is a plain Cantor bouquet displaying a

19

single fork at the neutral point N1=h(0, ee−1) = e. The set F[gc(z)] is simply the

complement of the set J[gc(z)] on the complex plane. When it happens tp= 1 with

p= 2, it also is t=−1 and the sequence of the inﬁnite exponential breaks into a

fork of period p= 2 on the complex plane, therefore the set J[gc(z)] consists of two

Cantor bouquets displaying a double fork at the neutral point N2=h(0, e−e) = 1/e.

The set F[gc(z)] then, consists of p= 2 sub-regions without attractors. Continuing

inductively, when it is tp= 1, p > 2, the sequence breaks into a fork of period p > 2

at the neutral point N3=h(0, c) = −W(−λ)/λ, therefore the set J[gc(z)] consists

of two Cantor bouquets displaying a p-furcation at the neutral point N3. The set

F[gc(z)] then, consists of p > 2 sub-cases without attractors.

•c∈[(RST )c∩Bδ(ϕ(t))]◦. Setting λ= log(c) = log(ϕ(t)) = t/et, we get the sub-

case |λ|=|t/et|with |t|>1 and tp= 1. If p≥2, then the case reduces to

sub-cases p≥2 of the cases 1 and 2, above. Consequently, if p > 1, the set

J[gc(z)] consists of two Cantor bouquets displaying a p-furcation at the repeller

R1=−W(λ)/λ and the set F[gc(z)] consists of psub-areas of the complex plane

(Fatou ﬂower in [78]), with each petal of the ﬂower containing the periodic attractor

Ap=g(k)(z0), z0= HW(. . . , λ)/λ,k∈ {0,1, . . . , p −1}, which leads any of the

points in the petals to Ap.

ϕ(t), which delineates the region RST , is conformal at every point of the unit circle

except at the point ϕ(1) = ee−1, therefore the three cases above cover fully the boundary

of RST except for a neighborhood Bδ(ϕ(1)), δ > 0, of the point ϕ(1), at which the map

ceases to be conformal. In any neighborhood of this point we can have |λ| ≥ 1/e, and in

such a case, the corresponding set J[gc(z)] suﬀers periodic explosions, which the references

call Knaster explosions, in which this set suddenly explodes and covers the entire complex

plane in the form of an indecomposable continuum. Otherwise, using δ > 0 we cover fully

the region RST \Bδ(ϕ(1)), therefore we know fully the sets J[gc(z)] and F[gc(z)] and the

proof of the proposition is complete.

4.3 General topological map of the inﬁnite exponential

A general map which describes the local behavior of the sequence αnon the complex plane

for the corresponding sets which will be created using a given base of the exponential

λ= log(c), for c∈Ccan be seen in [93] and [66] and is given here in Figure 3.

The colored regions are areas of period p≥1. The black pixels are points which have

escaped the size bound that has been set when imaging with the program. Consequently

the accuracy on these regions depends on the aforementioned escape bound set when

iterating the sequence of Euler.

The neutral points of J[g] are precisely the points ϕ(t), with tan n-th root of unity

and ϕ(z) = exp(z·exp(−z))), which lie in ∂RST .

The simplest case therefore is given by p= 1 and c=ee−1, which is a plain bouquet,

with one conspicuous neutral point at the vertex of the bouquet at h(0, c). The imme-

diately next case is given by p= 2 and c=e−e, which is two bouquets interacting at

the neutral point h(0, c). The general case for c=ϕ(t), with tp= 1, are two bouquets

which interact with period pat the neutral point h(0, c). The rest of the cases for |t|>1

give in general two bouquets which interact with period pat h(0, c) in combination with

20

Figure 3: Parameter map of gc(z), c∈C

the p-periodic attractors Ap, which are vertices of a deformed p-polygon on the complex

plane.

When a point moves from the interior of the region RST to the exterior, during its

passing from ∂RS T , we observe a periodic explosion or a fork of Cantor bouquets, in

which gradually show up pperiodic attractors around the point h(0, c), where pis the

pre-period of the multiplier of c, as in [76] and [78]. In other words, we observe a Knaster

explosion of period p.

The regions of the complex plane where αncauses forks of period pis precisely the

Fatou-like subset F[gc(z)] of the map of Figure 3. This subset consists of inﬁnitely thin

threads which divide the complex plane into regions where αnis periodic. These threads

create Cantor bouquets and their accuracy depends on the accuracy of the calculating

program. Had the accuracy of the calculations been unlimited, the threads of the bouquets

would not show at all, since the thickness of any single thread is actually zero.

Consequently, Lemma (3.3) generalizes the central Lemma (2.18) and classiﬁes nor-

mally the periods of the sequences for any point on the complex plane with sole exception

a neighborhood of ϕ(1). This answers Proposition (1.2) of Euler in its general form, that

is for general x=c̸=ee−1on the complex plane.

The characteristic points therefore of the topology of the inﬁnite exponential as it

follows from the above are exactly p+1 points: Complex inﬁnity, towards which move all

Cantor bouquet points and pattractors of the Fatou lake beds (when they exist) as they

are given by Lemma 3.3, towards which move all the points of these lake beds, in spiral

manner.

The torsion angle of the spiral convergence of the periodic attractors of the lake beds

is a function of the neighborhood of the point of the base of the exponential map in

relation to the phase angle of the boundary of the Shell-Thron region. This means that

21

to get spiral convergence with greater or smaller torsion angle, it suﬃces to change a bit

the angle of the pre-period of the base of the exponential map in a small neighborhood

of the base, relative to the previous phase angle of the base.

4.4 Topology

Proposition (4.2) shows that the topology of the set which results from the repeated

application of the exponential map, when it is not subject to Knaster explosions, is in

general a psymmetry or in other words, a Fatou ﬂower or diﬀerently, a deformed p-polygon

on the complex plane or otherwise a fork of Cantor bouquets of period p.

The topology of a Cantor bouquet is given in the references above and is homeomorphic

to the topology of a union of half open intervals [α, +∞), with the αof the bouquet vertices

in correspondence to the points of a sheared Cantor-like set CS (see section 4.5).

If the bouquet is CBh(k,c)=

k

[αk,+∞), in the general case, the topology of the

inﬁnite exponential, will be J[gc(z)] F[gc(z)] = CBh(k,c)× ⋄p. In other words, a fork

of Cantor bouquets, of period p.

Examples are given in Figures 5 and 8, 9, 10, 11, 12, 13 of the Appendix.

4.5 Fractal nature in the inﬁnite exponential

The Julia sets here, that is the Cantor bouquets of the inﬁnite exponential display an

intense fractal character. Speciﬁcally, for given screen resolution j, which is a function of

the bail-out value in the example of the code in the Appendix, Devaney represents them

using “ﬁngers” δj

n,n∈N.

We note a periodic break-up of each ﬁnger δjinto a subsequence of ﬁngers δj+1

n,n∈N,

for each apparent ﬁnger δj, which repeats ad inﬁnitum for each new ﬁnger. Consequently

the following relation gives a partial cover of the threads of a bouquet at resolution level

j, as δj=∞

n=0 δj+1

n. An example is given in Figure 4.

Figure 4: Devaney “ﬁnger” δjof Cantor bouquet

In an orthocentrically oriented ﬁnger δj, like that of ﬁgure 4 (oriented in this case with

inﬁnity down), the vertical width of the bouquet is bounded (see again [76], [75] and [77])

and the break up of half the ﬁnger can be set to a homeomorphic correspondence with

a suitable Cantor set, on the closed vertical interval [a, b], with athe vertical projection

22

of the vertex of δjon the real axis and bhalf the horizontal dimension of the ﬁnger.

The central lines in Figure 4 show approximately the the homeomorphic correspondence

implied for ﬁnger δj.

The dimension on either axis is additive, so if anand bn,n∈Nare the traces of the

bases of the (sub-)ﬁngers δj+1 on the interval [a, b], we write the recursive expression that

describes the break up of ﬁnger δjinto ﬁngers δj+1 as,

dimH,x(δj)∼∞

j=0

dimH(δj+1)x=∞

n=0 |bn−an|(13)

The total thickness of the ﬁnger is ﬁnite (2 |b−a|) in the horizontal dimension, there-

fore the sequence dimH,x(δn)∼n

j=0 |bn−an|is geometric with ratio,

r=

dimH,x(δn+1 )

dimH,x(δn)=δn+1 x

|δn|x

(14)

Note that ris independent of n(being the ﬁxed ratio of a geometric), therefore

equation (14) must be satisﬁed also for n= 1, which reveals that in the given resolution

of the ﬁnger δjwe have in our screen it suﬃces to measure the diﬀerence in the diameters

of any two successive relative to resolution level ﬁngers in the image of the bouquet.

If we had then at our disposal a “straight” bouquet against at least one dimension x

or y, then it would suﬃces to measure the diﬀerence in diameter between any two such

ﬁngers in screen pixels and we would immediately get the ratio of the geometric that is

the break up ratio of the Cantor bouquet by at least one direction.

There exists one and only one such bouquet. The bouquet produced by the exponential

map with base λ=e−1, which sees at point exp(1) (preperiod p= 1, center), which

includes the interval [e, +∞). For any other base apart from this lambda the behavior

of the exponential for a base in the right or left neighborhood of this point (λ) in the

direction θis unpredictable with Knaster explosions which deform the bouquets and force

them to suddenly cover the entire complex plane. Such a measuring of diﬀerence between

ﬁngers therefore is not possible in images of bouquets for base |λ|>1/e.

Yet we also have the images from all the other values of the base we calculated. No

bouquet however allows accurate measuring by speciﬁc dimension, because the bouquet

threads may suddenly turn unexpectedly towards a direction other than the expected one

we measure in, therefore any such measuring in this direction will necessarily produce

errors.

This way, the only bouquet available is that of λ= 1/e, since it’s the only one among

the one’s we’ve described, for which we know that exactly one of its threads is a straight

line. An image of this bouquet is given in Figure 5. The image of this bouquet however

extends all the way to positive inﬁnity (case of angle θ= 0), therefore it distorts the ratio

rof the geometric at least by direction θ.

The actual ratio of the geometric therefore can be measured in direction x≥ewith

the help of the map µ(z) = 1/z, ﬁnding the inverse image relative to inﬁnity as a fractal

set using some Maple code. It is the image which shows in Figure 6.

µis meromorphic in any neighborhood of inﬁnity |z| ≥ R > 0 and the exponential

with base λ=e−1is also meromorphic on the interval [e, +∞), therefore the image of

Figure 6 is conformal on [µ(R), µ(e)].

23

Figure 5: Cantor bouquet at c=e(of base λ=e−1)

Figure 6: Cantor bouquet at c=eunder the map µ(z) = 1/z

Therefore on this interval the dimension is given by the topology of the measure of the

absolute value of a geometric with successive ﬁnger intervals an+1 =bn, for each n∈N,

as

24

|µ(e)−µ(R)|x=∞

n=0

dimH,x(δn)

=∞

n=0 |δn|x

=∞

n=0 |bn−an|

=∞

n=0 |b0−a0| · rn=|b0−a0| · 1

1−r

(15)

Considering now a sequence Rn→ ∞, the measure of the geometric can be measured

experimentally from Figure 6, either as,

rx= lim

Rn→∞ 1−δ0x

|µ(e)−µ(Rn)|<1 (16)

or as,

rx=dimH,x(δ2)

dimH,x(δ1)=δ2x

|δ1|x

<1 (17)

Using equation (17) in2Figure 6, we ﬁnd,

rx∼

490 −337

337 −150= 0.8181 (18)

In the horizontal direction we therefore have dimH,x(δ)∼rx. In the vertical direction

(y) the dimension of the bouquet is at least that of the continuum ([e, +∞)) therefore,

1≤dimH,y(δ)

0≤dimH,x(δ)≤rx⇒

1≤dimH(δ)≤1 + rx= 1.818

We ﬁnally get 1 ≤dimH(CBe)≤1 + rx= 1.8181, which means3that the dimension

of the Cantor bouquet at e(λ=e−1) is fractional with geometric ratio of decomposition

∼rby at least one arbitrary direction.

Against the vertical direction then, the intersection of the bouquet λ= 1/e is home-

omorphic to a generalized4Cantor set, of dimension r, which is a fractal. The dimension

of such sets is analyzed partly in [60] and is given as a function of a general parameter

0< γ < 1, as

2For the measuring a ﬁgure of greater resolution was used.

3The java program on page “http://www.stevec.org/fracdim/” on the net gives r∼1.85 using the

box counting topology.

4Generalized but symmetric Cantor set, as the bouquet is symmetric relative to the axis [e, +∞).

25

r=ln(2)

ln(1−γ

2)(19)

In our case we have then r∼0.8181, therefore if we solve equation (17) for γ, we get

γ∼1/7, which means that the corresponding sheared set is homeomorphic to a Cantor

set which has a geometric break down ratio 1/7. The common known Cantor set has a

break down ratio equal to 1/3. We can therefore say that the bouquet at efor λ= 1/e

is a fractal. It is dense by vertical section and the set CBe{+∞} is connected. If we

remove however the point {+∞}, then the set is totally disconnected.

4.6 The primitive of a Cantor bouquet

The primitive of a Cantor bouquet therefore may be revealed if we examine the image

of the n-th roots of the neighborhood Bδ(ϕ(1)), under the map gc(z) and is presented

here in Figure 7 for n= 100, of the neighborhood δ= 0.01, of ϕ(1) = ee−1. The Figure

validates the aforementioned references which characterize it as homeomorphic to a C∞

brush.

Figure 7: Primitive of Cantor bouquet at c=e1/e

We conclude immediately that in as far as the bouquet for λ= 1/e is concerned, all

the threads of this bouquet are C∞curves on the complex plane.

The form of the primitive in combination to the measure of the fractal dimension,

reveals fully the structure of the bouquet on the microscope. What is visible elementarily

is in general a smooth (inﬁnitely diﬀerentiable) curve-thread, which periodically explodes

into a continuum of other inﬁnitely smooth curves-threads.

The noteworthy characteristic with the threads of any bouquet then, is that only the

coordinates of the thread [e, +∞) of the bouquet with λ= 1/e are known. Any other

thread is (computationally) unknown, meaning that no point (of any other thread) can

be determined exactly5, except the points towards which the bouquets “point” when we

5Except as a function of the resolution level of the screen jor equivalently of the bail-out value in the

code of the Appendix

26

have a bouquet fork of period p≥1. In contrast with the previous characteristic, the

attractors and the repellers of the Fatou lake beds can always be expressed analytically.

4.7 Pinball with the inﬁnite exponential

We also observe that the whole topology of the inﬁnite exponential is additionally periodic

modulo 0 < θ ≤2πon the complex plane, as all ﬁgures display bouquets which protrude

against the center under a speciﬁc angle θ. This angle is characteristic of the angle of the

pre-period of the base λof the exponential.

The previous periodicity allows us to concentrate on the topology of the Euler sequence

on the complex plane modulo one turn θwhich is ﬁnally equivalent to the dynamics of the

orbits of a ball in a pinball game on the strip {z:|ℜ(z)| ≤ bθ} × {−∞}, equipped with

the topology of the Fatou lake beds and highlands or Julia crevices with metric the speed

of convergence/attraction (also see [87]) or repulsion to inﬁnity and with the attractor at

inﬁnity being equivalent to gravity on a pinball table of angle θ.

Whatever is close to a Cantor bouquet sooner or later goes to complex inﬁnity (side

direction of gravity under plane angle θrelative to horizontal plane), while whatever

is inside a lake bed ﬁnally falls inside a small hole like in a regular pinball machine of

inclination θ, with plocal holes or dingy surface ornaments.

With the exception of a neighborhood of the point λ= log(c) = e−1, for rational

multiples of the pre-period angle 2π/p of the base λof the exponential map gc(z) = cz,

the topology of the inﬁnite exponential is (at least) a Cantor bouquet at a p-furcation in

a Fatou lake bed with center α, with αand p∈Ndetermined analytically.

5 The generalized inﬁnite exponential

5.1 Deﬁnitions

The general inﬁnite exponential is analyzed in a limited way in [210] ([18],[178], [179])

and in [122]. It is deﬁned recursively:

Deﬁnition 5.1 If Zk={z1, z2, . . . , zk},k∈N, with zk∈C\ {x∈R:x≤0}and n≤

|Zk|=k,

en(Zk) = zk, if n= 1,

zen−1(Zk)

k−n+1 , if n > 1.

Whenever we need to study convergence of the previous exponential, we simply ex-

amine the values of the sequence bn=en(Zn). Unwinding recursively bn,n∈N, using

Deﬁnition (5.1), produces the terms: z1, zz2

1, zzz3

2

1, . . . , zzz···zn

3

2

1.

There is however a dual deﬁnition which reverses the indexes, as

Deﬁnition 5.2 Given a list Zk={z1, z2, . . . , zk},k∈N, with zk∈C\ {x∈R:x≤0}

and n≤ |Zk|=k,

e∗

n(Zk) = z1, if n= 1,

ze∗

n−1(Zk)

n, if n > 1.

27

Analyzing recursively the sequence b∗

n=e∗

n(Zn), n∈N, using Deﬁnition (5.2), pro-

duces the sequence z1, zz1

2, zzz1

2

3, . . . , zzz···z1

n−2

n−1

n.

Whenever the general inﬁnite exponential converges, we write,

Deﬁnition 5.3 Whenever the following limit exists ﬁnitely,

e∞(Z∞) = lim

n→∞ en(Zn) = lim

n→∞ bn=b∞

e∗

∞(Z∞) = lim

n→∞ e∗

n(Zn) = lim

n→∞ b∗

n=b∗

∞

The references ([18]) show the following (in relation to Deﬁnitions (5.1) and (5.2)):

Theorem 5.4 (Barrow) The sequence bnconverges, if and only if:

• ∃n0:∀n≥n0:bn∈[(1/e)e, e(1/e)].

•znconverges.

Theorem (5.4) can be now generalized, since we now know the full extent of the RST

region:

Theorem 5.5 (Barrow) Sequence bnconverges, if and only if:

• ∃n0:∀n≥n0:bn∈RST .

•znconverges.

We are interested in ﬁnding interesting expressions for the limits b∞and b∗

∞.

5.2 Ascending indexes

For the ﬁrst case (Deﬁnition (5.1)), we set Gn(z) = e∗

n+1(Ln∪ {z}), in which case we get

the iteration:

Gn(z) = y⇒

log(Gn(z)) = log(y)⇒

Gn−1(z) = log(y)

log(zn)⇒

log(Gn−1(z)) = log log(y)

log(zn)⇒

Gn−2(z) = log(Gn−1(z))

log(zn−1)⇒

·· ·

Gn−k+1(z) = log(Gn−k(z))

log(zn−k)

Gm+1(z) = log(Gm(z))

log(zm)

(20)

28

Suppose z0= lim

n→∞ zn. We select ksuch that m=n−k≥n0. We can take limits for

m→ ∞ in the last of (20) and solve for W:

G(z) = log(G(z))

log(z0)⇒

G(z) = W(−log(z0))

−log(z0)=h(z0)

(21)

Equations (21) show,

Theorem 5.6 If z0= lim

n→∞ zn, then for ϵ > 0, there exists n0, such that for n≥n0,

|b∗

n−h(z0)|< ϵ.

Proof: For any ϵ > 0, we can ﬁnd n0suﬃciently big that guarantees |zn−z0|< ϵ and

then the bases of the tower until zn0can be expressed as z0+ϵ. The elements further

than zn0can be gradually sent to inﬁnity, by choosing big n0and then the limit of the

tower will approach the number h(z0+ϵ). The Theorem follows by forcing ϵ→0.

5.3 Descending indexes

For the dual case (Deﬁnition (5.2)), if Zk={z1, z2, . . . , zk}, transforming as wm=

zn−m+1 and as Z∗

k={w1, w2, . . . , wk}to change the order of the indexes, we set Gn(z) =

e∗

n(Z∗

n∪ {z}), in which case we get the iteration:

Gn(z) = y⇒

log(Gn(z)) = log(y)⇒

Gn−1(z) = log(y)

log(wn)⇒

log(Gn−1(z)) = log log(y)

log(wn)⇒

Gn−2(z) = log(Gn−1(z))

log(wn−1)⇒

·· ·

Gn−k+1(z) = log(Gn−k(z))

log(wn−k)

Gm+1(z) = log(Gm(z))

log(wm)

(22)

Suppose z0= lim

n→∞ zn. We choose ksuch that m=n−k≥n0. Similarly, we can get

the limit for m→ ∞ in the last of (22) and to solve for W:

G(z) = log(G(z))

log(z0)⇒

G(z) = W(−log(z0))

−log(z0)=h(z0)

(23)

29

Equations (23) show,

Theorem 5.7 If z0= lim

n→∞ zn, for each ϵ > 0, there exists n0, such that for each n≥n0,

|bn−en0+1(Zn0∪ {h(z0)})|< ϵ.

Proof: Similarly, given ϵ > 0, we can ﬁnd n0big enough that guarantees |zn−z0|< ϵ,

and then all the terms higher than zn0can be approximated as h(z0+ϵ). The tower below

zn0, leaves the elements untouched, therefore the approximation will be zz···

zh(z0)+ϵ

n0

2

1. The

Theorem follows by forcing ϵ→0.

Theorems (5.6) and (5.7) in combination with Proposition (4.2) answer Proposition

(1.2) of Euler in its general form. Speciﬁcally,

Corollary 5.8 Suppose the sequence γn,n∈Nwith γn→c∈C. The behavior of the

sequence αnwith αn+1 =γαn

n,n∈N, is described by the sets J[gc+ϵ]and F[gc+ϵ], with

ϵ > 0from Theorem (5.6).

30

6 Appendix: programming/results

Executable Maple code is attached in a separate .pdf document, to which the author refers

for the validation of the Corollaries and the creation of the Figures of the dissertation. In

the ﬁgures, the ﬁrst column contains an attractor A1=h(0, c), the second column contains

the neutral point N1=h(0, c) and the third column contains the repeller R1=h(0, c)

and the periodic attractors Apof Proposition (4.2), distributed around the repeller R1.

Figure 8: Preperiod p= 1, with λ= log(c) = t/et,|t|<1, t= 1 and |λ|>1/e

Figure 9: Preperiod p= 2, with λ= log(c) = t/et,|t|<1, t=eπ i =−1 and |t|>1

Figure 10: Preperiod p= 3, with λ= log(c) = t/et,|t|<1, t=e2π/3iand |t|>1

31

Figure 11: Preperiod p= 4, with λ= log(c) = t/et,|t|<1, t=eπ /2iand |t|>1

Figure 12: Preperiod p= 5, with λ= log(c) = t/et,|t|<1, t=e2π/5iand |t|>1

Figure 13: Preperiod p=π, with λ= log(c) = t/et,|t|<1, t=e2iand |t|>1

32

7 Acknowledgements

The author wishes to thank all the factors who contributed in the exposition of the work.

Speciﬁcally, his advisor and professor Yiannis Papadoperakis for the almost unbeliev-

able patience he showed during the routing of the dissertation, professor Paris Pamﬁlos

and professor Michalis Lambrou for their recommendation letters and all the factors that

solved many subproblems, like the researcher Alexandr Dubinov who solved Kepler’s

equation, the professors of the University of Crete, the advisors and mathematicians of

the internet who answered many of the questions in international forums and usenet,

like professor Robert B. Israel et al., the mathematics programmers of the Maple forum

et al., Leroy Quet who solved the problem of the series development of the exponential

(Lemma (8.1) in [210]), Daniel Geisler of the page tetration.org, Henryk Trappman, An-

drew Robbins, Gottfried Helms et al., of the tetration forum, Eric Weisstein of the page

mathworld.wolfram.com the referees of the author’s articles, professor Dave L. Renfro who

collected, archived and sent most of the references and the creators of the used fractal

programs which validated the results of the work, like Jan Hubicka and Thomas Marsh.

More acknowledgements can be found in the author’s references. The author dedicates

the dissertation to his parents and to his ﬁrst mathematics professor Kostas Sourlas.

33

8 Bibliography

References

[1] W.B. Manhard 2d. Is exponentiation commutative? Mathematics Teacher,

74(1):56–60, Jan. 1981.

[2] 30 September 1905 8-th regular meeting of the San Franscisco Section of the AMS,

University of California. Comment 8. Bulletin of the American Mathematical Soci-

ety, 12:115, 1905-1906.

[3] D.S. Alexander. A History of Complex Dynamics, From Schroeder to Fatou and

Julia. Friedrich Vieweg and Sohn Verlagsgesellschaft mbH, Oct. 1994.

[4] A.O. Allen. eπor πe?Journal of Recreational Mathematics, 2(4):255–256, October

1969.

[5] M.M. Alliaume. Sure le developpment en serie de la racine d’ une equation. Mathesis

Requeil Mathematique, 44:163–166, 1930.

[6] J.C. Appleby. Notes on hexponentiation. The Mathematical Gazette, 79(484):x–y,

1995.

[7] R.C. Archibald and R. Clare. Problem 136. Mathematics of Computation, 6(39):204,

July 1952.

[8] F. Arjomandi. Problem 27.1. Mathematical Spectrum, 27(3):68, 1994-1995.

[9] J.M. Ash. The limit of xx···x

as x tends to zero. Mathematics Magazine, 69:207–209,

1996.

[10] A.L. Baker. Functional exponents. School Science and Mathematics, 8(3):225–227,

Mar. 1908.

[11] I.N. Baker and P.J. Rippon. Convergence of inﬁnite exponentials. Annales

Academiae Scientiarum Fennicae. Series A AI Mathematics, pages 179–186, 1983.

[12] I.N. Baker and P.J. Rippon. A note on complex iteration. American Mathematical

Monthly, 92:501–504, 1985.

[13] I.N. Baker and P.J. Rippon. A note on inﬁnite exponentials. The Fibonacci Quar-

terly, 23:106–112, 1985.

[14] I.N. Baker and P.J. Rippon. Iterating exponential functions with cyclic expo-

nents. Mathematical Proceedings of the Cambridge Philosophical Society, 105:359–

375, 1989.

[15] I.N. Baker and P.J. Rippon. Towers of exponents and other composite maps. Com-

plex Variables, 12:181–200, 1989.

[16] E. Barbette. Des progressions logarithmiques. Mathesis Requeil Mathematique,

8(2):135–137, 1898.

34

[17] M. Barnsley. Fractals Everywhere. Academic Press, Inc., Cambridge, MA., 1988.

[18] D.F. Barrow. Inﬁnite exponentials. American Mathematical Monthly, 43:150–160,

1936.

[19] K.-H. Becker and M. D¨orﬂer. Dynamical Systems and Fractals. Cambridge Univer-

sity Press, 1990.

[20] S. Bedding and K. Briggs. Iteration of quaternion functions. American Mathematical

Monthly, 8(103):654–664, Oct. 1996.

[21] E.T. Bell. Iterated exponential numbers. Bulletin of the American Mathematical

Society, 43:774–775, 1937.

[22] E.T. Bell. The iterated exponential integers. The Annals of Mathematics, Second

Series, 39:539–557, 1938.

[23] W.W. Beman. An inﬁnite exponential. Mathematical Spectrum, 27(1):22, 1994-

1995.

[24] W.W. Beman. Problem 389. American Mathematical Monthly, 21(1):23, January

1914.

[25] C.M. Bender. Advanced Mathematical Methods for Scientists and Engineers, Ap-

proximate Solution of Nonlinear Diﬀerential Equations. McGraw Hill, 1978.

[26] A.A. Bennet. The iteration of functions of one variable. The Annals of Mathematics,

Second Series, 17:23–60, 1915.

[27] A.A. Bennet. Note on an operation of the third grade. The Annals of Mathematics,

Second Series, 17:74–75, 1915.

[28] A. Bennett. Note on an operation of the third grade. The Annals of Mathematics,

Second Series, 17(2):74–75, Dec. 1915.

[29] A. Bennett. The iteration of functions of one variable. The Annals of Mathematics,

Second Series, 17(1):23–60, Sep. 1915.

[30] W. Bergweiler. Periodic points of entire functions. Complex Variables Theory Appl.,

17:57–72, 1991.

[31] W. Bergweiler. Iteration of meromorphic functions. Bulletin of the American Math-

ematical Society, 29:151–188, 1993.

[32] B.C. Berndt. Ramanujan’s Quarterly Reports. Bull. London Math. Soc.,

16(1984):469–489, Apr. 1983.

[33] H.J.M. Bos. Johann Bernoulli on exponential curves, ca. 1695 innovation and ha-

bituation in the transition from explicit constructions to implicit functions. Nieuw

Archief voor WisKunde, 14(4):1–19, 1996.

[34] L. Brand. Binomial expansions in factorial powers. American Mathematical

Monthly, 67(10):953–957, Dec. 1960.

35

[35] L. Brand. The roots of a quaternion. The American Mathematical Monthly,

49(8):519–520, Oct. 1942.

[36] B. Branner. Holomorphic dynamical systems in the complex plane an introduction,

1995. Proceedings of the 7th EWM meeting, Madrid.

[37] N. Bromer. Superexponentiation. Mathematics Magazine, 60:169–174, 1987.

[38] H.J. Brothers and J.A. Knox. New closed-form approximations to the logarithmic

constant e.Mathematical Intelligencer, 20(4):25–29, Fall 1998.

[39] M. Brozinsky. Problem 4394. School Science and Mathematics, 93(6):340, Oct.

1993.

[40] B.W. Brunson. The partial order of iterated exponentials. American Mathematical

Monthly, 93:779–786, 1986.

[41] E.P. Bugdanoﬀ. Problem 3515 [1931,462]. American Mathematical Monthly,

39(9):552–555, Nov. 1932.

[42] A.P. Bulanov. Iterated cyclic exponentials and power functions with extra-periodic

ﬁrst coeﬃcients. Sb. Math, 201(1):23:23–55, 2010.

[43] L.E. Bush. The 1961 william lowell putnam mathematical competition. American

Mathematical Monthly, 69:759–767, 1962.

[44] F.A. Butter. A note on the equation xy=yx.American Mathematical Monthly,

46(6):316–317, June/July 1939.

[45] W.D. Cairns. Problem 422. American Mathematical Monthly, 22(4):133, Apr. 1915.

[46] F. Cajori. History of the exponential and logarithmic concepts (part 4). American

Mathematical Monthly, 20(4):107–117, Apr. 1913.

[47] F. Cajori. History of the exponential and logarithmic concepts (part 2). American

Mathematical Monthly, 20(2):35–37, Feb. 1913.

[48] F. Cajori. History of the exponential and logarithmic concepts. American Mathe-

matical Monthly, 20(1):5–14, Jan. 1913.

[49] F. Cajori. History of the exponential and logarithmic concepts (part 6). American

Mathematical Monthly, 20(6):173–182, June 1913.

[50] F. Cajori. Errata in the february issue. American Mathematical Monthly, 20(3):104,

Mar. 1913.

[51] F. Cajori. History of the exponential and logarithmic concepts (part 3). American

Mathematical Monthly, 20(3):75–84, Mar. 1913.

[52] F. Cajori. Napier’s logarithmic concept. American Mathematical Monthly, 23(3):71–

72, Mar. 1916.

36

[53] F. Cajori. History of the exponential and logarithmic concepts (part 5). American

Mathematical Monthly, 20(5):148–151, May. 1913.

[54] F. Cajori. History of the exponential and logarithmic concepts (part 7). American

Mathematical Monthly, 20(7):205–210, Sep. 1913.

[55] R.D. Carmichael. On a certain class of curves given by transcendental equations.

American Mathematical Monthly, 13:221–226, 1906.

[56] R.D. Carmichael. On certain transcendental functions deﬁned by a symbolic equa-

tion. American Mathematical Monthly, 15:78–83, 1908.

[57] R.D. Carmichael. Problem 275. American Mathematical Monthly, 14(2):27, Feb.

1907.

[58] G.F. Carrier, M. Krook, and C.E. Pearson. Functions of a Complex Variable.

McGraw-Hill Book Company, New York, 1966.

[59] H.S. Carslaw. Relating to Napier’s logarithmic concept. American Mathematical

Monthly, 23(8):310–315, Oct. 1913. (Comments on the Cajori article and a response

by professor Cajori).

[60] A.Yu. Cherny, E.M. Anitas, A.I. Kuklin, M. Balasoiu, and V.A. Osipov. Scattering

from generalized cantor fractals. http://arxiv.org/abs/0911.2497.

[61] R.V. Churchill, J.W. Brown, and R.F. Verhey. Complex Variables and Applications.

McGraw-Hill, New York, 1974.

[62] Jr. C.J. Everett. An exponential diophantine equation. American Mathematical

Monthly, 43(4):229–230, Apr. 1936.

[63] P. Colwell. Bessel functions and Kepler’s equation. American Mathematical

Monthly, 99(1):45–48, Jan. 1992.

[64] S.D. Conte and Carl de Boor. Elementary Numerical Analysis. McGraw-Hill Book

Company, New York, 1980.

[65] R. Cooper. The relative growth of some rapidly increasing sequences. Journal of

the London Mathematical Society, 29(1):59–62, 1954.

[66] R.M. Corless. Lambert W Function, 2000.

http://www.apmaths.uwo.ca/rcorless/frames/PAPERS/LambertW.

[67] R.M. Corless, G.H. Gonnet, D.E. Hare, D.J. Jeﬀrey, and D.E. Knuth. On the

Lambert W function. Advances in Computational Mathematics, 5:329–359, 1996.

[68] R.M. Corless, D.J. Jeﬀrey, and D.E. Knuth. A sequence of series for the Lambert

W function. Proceedings of the 1997 International Symposium on Symbolic and

Algebraic Computation, Maui, Hawaii. New York: ACM Press, pages 197–204,

1997.

37

[69] Alex D.D. Craik. Prehistory of Fa´a Di Bruno’s formula. American Mathematical

Monthly, 112(2):217–234, Feb. 2005.

[70] R.E. Crandall and M.M. Colgrove. Scientiﬁc Programming with Macintosh Pascal.

Wiley Press, 1986.

[71] C.C. Cross. Problem 93. American Mathematical Monthly, 6(8/9):199–201,

Aug./Sep. 1899.

[72] P.J. Davis. Leohard Euler’s integral: A historical proﬁle of the Gamma function.

American Mathematical Monthly, 66(10):849–869, Dec. 1959.

[73] C. A. Deavours. The quaternion calculus. The American Mathematical Monthly,

9(80):995–1008, 11 Nov., 1973.

[74] E. Delville. Nombres egaux a leurs logarithmes. Mathesis, 4(2):264, 1912.

[75] R.L. Devaney. Accessible points in the julia sets of stable exponentials. Discrete

and Continuous Dynamical Systems, 1:299–318, 2001.

[76] R.L. Devaney. Cantor bouquets, explosions and knaster continua: Dynamics of

complex exponentials. Topology and its Applications, 110:133–161, 2001.

[77] R.L. Devaney. Cantor and Sierpinski, Julia and Fatou: Complex topology meets

Complex dynamics. Notices of the American Mathematical Society, 51(1):9–15,

2004.

[78] R.L. Devaney. Complex Exponential Dynamics (in Handbook of Dynamical Sys-

tems), Vol. 3. Eds. H. Broer, F. Takens, B. Hasselblatt, 2010.

[79] A.C. Dixon. On B¨urmann’s theorem. Proc. London Math. Soc., 34:151–153, 1902.

[80] A.E. Dubinov, I.D. Dubinova, and S.K. Saikov. Lambert W function: Table of Inte-

grals and Other Mathematical Properties [in Russian]. Sarov Physical and Technical

Institute, Sarov, 2004.

[81] R.M. Dudley. Problem 5098 [1963,445]. American Mathematical Monthly, 71(5):563,

May. 1964.

[82] Jr. A. M. E.D.Roe. Problem 68. American Mathematical Monthly, 5(4):110, Apr.

1898.

[83] G. Eisenstein. Entwicklund von aaa...

.Journal Fur Die Reine Und Angewandte

Mathematik, 28(8):49–52, 1944.

[84] G. Enestrom. Es ware ratsam, den passus. Bibliotheca Mathematica, 13(3):270,

1912-1913.

[85] L. Euler. De formulis exponentialibus replicatus. Acta Sci. Petropol., 1:38–60, 1778.

[86] C.L. Evans. Interesting logs, reader’s reﬂection column. Mathematics Teacher,

72(7):489, Oct. 1979.

38

[87] L. Han F. Gao and K. Schilling. On the rate of convergence of iterated exponentials.

Journal of Applied Mathematics and Computing, pages 1–8, Sep. 2011.

[88] M.J. Farkas. Sur le fonctions iteratives. Journal de Mathematiques Pures et Ap-

pliquees [=Liouville’s Journal], 10(3):101–108, 1884.

[89] P. Franklin. Relating to the real locus deﬁned by the equation xy=yx.American

Mathematical Society, 24:137, 1917.

[90] J. Friedman. On the Convergence of Newton’s Method. PhD thesis, University of

California, Berkeley, 1987.

[91] B. Frizel. The problem of deﬁning the set of real numbers. Bulletin of the American

Mathematical Society [28-th regular meeting of the Chicago section of the AMS,

University of Chicago, 28-30 December 1910]., 17:296, 1910-1911.

[92] R. Garver. On the approximate solution of certain equations. American Mathemat-

ical Monthly, 39(8):476–478, Oct. 1932.

[93] D. Geisler. The Atlas of Tetration, 2004. http://www.tetration.org/.

[94] G.M. Ghunaym. Problem 4696. School Science and Mathematics, 99(1):54, Jan.

1999.

[95] R.A. Gibbs. Problem 4334. School Science and Mathematics, 92(5):292–293,

May/Jun. 1992.

[96] J. Ginsburg. Iterated exponentials. Scripta Mathematica II, pages 292–293, 1945.

[97] J.W.L. Glaisher. A theorem in trigonometry. Quarterly Journal of Pure and Applied

Mathematics, 15:151–157, 1878.

[98] F. Gobel and R.P. Nederpelt. The number of numerical outcomes of iterated powers.

American Mathematical Monthly, 78:1097–1103, 1971.

[99] R.L. Goodstein. The equation ab=ba.The Mathematical Gazette, 28(279):76, May

1944.

[100] R. Greenberg. Problem e 1597 [1963, 757]. American Mathematical Monthly,

71(3):322, Mar. 1964.

[101] M.J.-H. Grillet. Les exponentielles successives d’Euler. Journal de Mathematiques

Pures et Appliquees [=Liousville’s Journal], 10:233–241, 1985.

[102] Sir William Rowan Hamilton. On quaternions. Proceedings of the Royal Irish

Academy, 3(1847):1–16, 11 Nov., 1844.

[103] Sir William Rowan Hamilton. On a new species of imaginary quantities connected

with a theory of quaternions. Proceedings of the Royal Irish Academy, 2:424–434,

13 Nov., 1843.

[104] D. Hammer. Problem 171, parenthetical roots. Mathematics and Computer Edu-

cation, 17(1):73, Winter 1983.

39

[105] M.E. Haruta. Newton’s method on the complex exponential function. Transactions

of the American Mathematical Society, 351(5):2499–2513, 1999.

[106] A. Hathaway. Quaternion space. Transactions of the American Mathematical So-

ciety, 1(3):46–59, Jan. 1902.

[107] A. Hausner. Algebraic number ﬁelds and the diophantine equation mn=nm.

American Mathematical Monthly, 68:856–861, 1961.

[108] A. Hausner. Problem e 1474 [1961,573], the equation mnm=nmn.American

Mathematical Monthly, 69(2):169, Feb. 1962.

[109] N.D. Hayes. The roots of the equation x= (c·exp)nxand the cycles of the

substitution (x|cex). Quarterly Journal of Mathematics (Oxford), 3(2):81–90, 1952.

[110] J. Van Heijenoort. From Frege to Goedel: A Source Book in Mathematical Logic,

1879 - 1931. Section On Hilbert’s construction of the real numbers, written by Wil-

helm Ackermann (1928). Harvard University Press, 1967.

[111] A.S. Hendler. Problem e 1144 [1954,711]. American Mathematical Monthly,

62(6):446, Jun./Jul. 1955.

[112] J.L. Hickman. Analysis of an exponential equation with ordinal variables. Proc-

ceedings of the American Mathematical Society, 61:105–111, 1976.

[113] A. Hoorfar and M. Hassani. Inequalities of the Lambert W function and hyperpower

function. Journal of Inequalities in pure and applied mathematics, 9(2):51–56, 2008.

[114] S. Hurwitz. On the rational solutions of mn=nm, with m/nen.American Mathe-

matical Society, 74:298–300, 1967.

[115] Jr. H.W. Labbers. Problem 420. Nieuw Archief voor WisKunde, 3(24):207–210,

1976.

[116] J.T. Varner III. Comparing ab=bausing elementary calculus. Two Year College

Mathematics Journal, 7(4):46, 1976.

[117] R.B. Israel. Calculus the Maple Way. Addison-Wesley, 1996.

[118] V.F. Ivanoﬀ. Problem e 34 [1933,241]. American Mathematical Monthly, 44(2):101–

106, Feb. 1934.

[119] S. J. J. G. Hagen. On the history of the extensions of the calculus. Bulletin of the

American Mathematical Society, 6:381–390, 1899-00.

[120] D.J. Jeﬀrey. The multi-valued nature of inverse functions, 2001. Available online

at: http://citeseer.ist.psu.edu/jeﬀrey01multivalued.html.

[121] H.W. Lenstra Jr. 3053. problem 566. Nieuw Archief voor WisKunde, 3(28):300–302,

1980.

[122] R.A. Knoebel. Exponentials reiterated. American Mathematical Monthly, 88:235–

252, 1981.

40

[123] M. Kossler. On the zeros of analytic functions. Proceedings of the London Mathe-

matical Society, 19(2):back pages for the January 13, 1921 Meeting, 1921.

[124] S.G. Krantz. Handbook of Complex Variables. Birkh¨auser, Boston, MA, 1999.

[125] Y.S. Kupitz and H. Martini. On the equation xy=yx.Elemente der Mathematik,

55:95–101, 2000.

[126] L.J. Lander. Problem e 1124 [1954,423]. American Mathematical Monthly,

62(2):124–125, Feb. 1955.

[127] H. L¨anger. An elementary proof of the convergence of iterated exponentials. Ele-

mente der Mathematik, 51:75–77, 1996.

[128] H. Langer. An elementary proof of the convergence of iterated exponentials. Ele-

mente Math., 51:75–77, 1996.

[129] J. Lense. Problem 3053. American Mathematical Monthly, 31:500–501, 1924.

[130] S. De Leo and P. Rotelli. A new deﬁnition of hypercomplex analyticity, January

1997. Dipartimento di Fisica, Universit`a degli Studi Lecce INFN, Sezione di Lecce

via Arnesano, CP 193, I-73100 Lecce, Italy, Dipartimento di Metodi e Modelli

Matematici per le Scienze Applicate, via Belzoni 7, I-35131 Padova, Italy.

[131] S.L. Loney. Plane Trigonometry, 5’th Edition. Cambridge University Press, parts

I and II, 1925.

[132] G.F. Lowerre. A logarithm problem and how it grew. The Mathematics Teacher,

72:227–229, 1979.

[133] S.V. L¨udkovsky and F. van Oystaeyen. Diﬀerentiable functions of quaternion vari-

ables. Bulletin des Sciences Mathematiques, 9(127):755–796, 2002.

[134] J. Van De Lune. Problem 899. Nieuw Archief voor WisKunde, 4(13):251–252, 1995.

[135] J. Macdonnell. Some critical points of the hyperpower function xx···x

.International

Journal of Mathematical Education, 20(2):297–305, 1989.

[136] A.J. Macintyre. Convergence of ii···.Procceedings of the American Mathematical

Society, 17:67, 1966.

[137] R.L. Mayes. Discovering relationships, logarithmic and exponential functions.

School Science and Mathematics, 94(7):367–370, Nov. 1994.

[138] J. Milnor. Dynamics in one complex variable introductory lectures. Institute for

Mathematical Sciences, SUNY, Stony Brook, New York, preprint 1990/5.

[139] M.C. Mitchelmore. A matter of deﬁnition. American Mathematical Monthly,

81:643–647, 1974.

[140] J. Morton. Graphical study of xx=y.The Mathematical Student Journal, 9(2):5,

Jan. 1962.

41

[141] E.J. Moulton. The real function deﬁned by: xy=yx.American Mathematical

Monthly, 23:233–237, 1916.

[142] R.P. Nederpelt. Elementary problems and solutions: E1903. American Mathemat-

ical Monthly, 79:395–396, 1972.

[143] D.J. Newman. Problem 4569 [1954,51]. The Mathematical Student Journal,

62(3):190–191, Mar. 1955.

[144] I. Niven. Which is larger, eπor πe?Two Year College Mathematics Journal,

3(2):13–15, 1972.

[145] S. Ocken. Convergence criteria for attracting cycles of Newton’s method. SIAM J.

Appl. Math., 58(1):235–244, 1998.

[146] C.S. Ogilvy. Advanced problems and solutions: E853. American Mathematical

Monthly, 56:555–556, 1949.

[147] L. O’Shaughnessy. Problem 433. American Mathematical Monthly, 26(1):37–39,

Jan. 1989.

[148] F.D. Parker. Integrals of inverse functions. The American Mathematical Monthly,

62(6):439–440, 1955.

[149] H.-O. Peitgen and P.H. Richter. The Beauty of Fractals. Springer-Verlag, Berlin,

Germany, 1986.

[150] J.J. Petersen and P. Hilton. exp(π)> πe?Readers Reﬂections Column, Mathemat-

ics Teacher, 74(7):501–502, Oct. 1981.

[151] H. Poritsky. Problem 2851 [1920,377]. American Mathematical Monthly, 29(3):132–

133, Mar. 1922.

[152] S. Rabinowitz. Problem 191, exponential roots. Mathematics and Computer Edu-

cation, 18(2):150–151, Spring 1984.

[153] W.R. Ransom. Problem e 3 [1932,489]. American Mathematical Monthly, 40(2):113,

Feb. 1933.

[154] D. Redfern. The Maple Handbook. Springer-Verlag, 1996.

[155] D.L. Renfro. Exponential and logarithmic commutativity. The Mathematics

Teacher, 91:275–362, 1998.

[156] J. Riordan. A note on Catalan parantheses. American Mathematical Monthly,

80:904–906, 1973.

[157] B. Ross and B.K. Sacheva. The inversion of Kepler’s equation. Int. J. Math. Educ.

Sci. Technol., 16:558–560, 1985.

[158] R.R. Rowe. The mutuabola. Journal of Recreational Mathematics, 3(3):176–178,

July 1970.

42

[159] S. Saks and A. Sygmund. Analytic Functions. Hafner Publishing Company, New

York, 1952.

[160] H. E. Salzer. An elementary note on powers of quaternions. American Mathematical

Monthly, 5(59):298–300, May 1952.

[161] D. Sato. Shorter notes: Algebraic solution of xy=yx, (0 < x < y). Procceedings of

The American Mathematical Society, 31:316, 1972.

[162] J. Scheﬀer. Problem 307. American Mathematical Monthly, 16(2):32, Feb. 1909.

[163] E.D. Schell. Problem e 640 [1944,472]. American Mathematical Monthly, 52(5):278–

279, May 1945.

[164] D.L. Shell. Convergence of Inﬁnite Exponentials. PhD thesis, University of Cincin-

nati, 1959.

[165] D.L. Shell. On the convergence of inﬁnite exponentials. Procceedings of The Amer-

ican Mathematical Society, 13:678–681, 1962.

[166] T.W. Shilgalis. Graphical solution of the equation ab=ba.The Mathematics

Teacher, 66:235, 1973.

[167] S.L. Siegel. Exponential equations. Reader Reﬂections Columnm, Mathematics

Teacher, 100(4):238, Nov. 2006.

[168] D.J. Silverman. What’s the limit? Journal of Recreational Mathematics, 4(2):144,

Apr. 1971.

[169] H.L. Slobin. The solutions of xy=yx,x, y > 0, x̸=yand their graphical repre-

sentation. American Mathematical Monthly, 38:444–447, 1931.

[170] G.P. Speck. abversus ba.School Science and Mathematics, 65(6):489–490, Jun.

1965.

[171] M. Spivak. Calculus (English). Publish or Perish, Inc., Berkeley, California, 1980.

[172] V.M. Spunar. Problem 430. American Mathematical Monthly, 26(9):415, Nov. 1919.

[173] R.M. Sternheimer. On certain integers which are obtained by repeated exponenti-

ation. Journal of Recreational Mathematics, 22(4):271–276, 1990.

[174] Daniel Drew (student). A recursion relation involving exponentials. American

Mathematical Monthly, 56(9):660–664, Nov. 1949.

[175] Paul Heckbert (student). |x|y=|y|x.The Mathematics Student Journal, 22(4):4,7,

Apr. 1975.

[176] A. Sudbery. Quaternionic analysis. Mathematical Procceedings of the Cambridge

Philosophical Society, 85:199–225, 1979.

[177] W.R. Thomas. John Napier. The Mathematical Gazette, 19(234):192–205, Jun.

1935.

43

[178] W.J. Thron. Convergence of inﬁnite exponentials with complex elements. Procceed-

ings of The American Mathematical Society, 8:1040–1043, 1957.

[179] W.J. Thron. Convergence regions for continued fractions and other inﬁnite pro-

cesses. American Mathematical Monthly, 68:734–750, 1961.

[180] S.D. Turner. Under what conditions can a number equal its logarithm? (part 2).

School Science and Mathematics, 28(240):376–379, Apr. 1928.

[181] S.D. Turner. Under what conditions can a number equal its logarithm? (part 1).

School Science and Mathematics, 27(7):750–751, Oct. 1927.

[182] H.S. Uhler. On the numerical value of ii.American Mathematical Monthly,

28(3):114–121, Mar. 1921.

[183] unknown. Problems, notes, 7-9. American Mathematical Monthly, 28(3):140–143,

March 1921.

[184] unknown. Problems for solution. American Mathematical Monthly, 70(5):571–573,

May 1963.

[185] G.T. Vickers. Experiments with inﬁnite exponents. Mathematical Spectrum,

27(2):34, 1994/1995.

[186] G.T. Vickers. More about an inﬁnite exponential. Mathematical Spectrum, 27(3):54–

56, 1994/1995.

[187] J.M. De Villiers and P.N. Robinson. The interval of convergence and limiting func-

tions of a hyperpower sequence. American Mathematical Monthly, 93:13–23, 1986.

[188] R. Voles. An exploration of hyperpower equations nx=ny.The Mathematical

Gazette, 83(497):210–215, 1999.

[189] D. Wang and F. Zhao. The theory of smale’s point estimation and its applications.

Journal of Computational and Applied Mathematics, 60:253–269, 1995.

[190] M. Ward. Note on the iteration of functions of one variable. Bulletin of the American

Mathematical Society, 40:688–690, 1934.

[191] M. Ward and F.B. Fuller. Continuous iteration of real functions. Bulletin of the

American Mathematical Society, 42:383–386, 1936.

[192] G. Waters. problem 185. Mathematics and Computer Education, 18(1):69–70, Win-

ter 1984.

[193] W.V. Webb. Rooting around for roots. Mathematics and Computer Education,

24(3):273, Fall 1990.

[194] Eric W. Weisstein. Conformal mapping, from mathworld–a wolfram web resource.,

2011. http://mathworld.wolfram.com/ConformalMapping.html.

[195] Eric W. Weisstein. Contour winding number, from mathworld–a wolfram web re-

source., 2011. http://mathworld.wolfram.com/ContourWindingNumber.html.

44

[196] F. Woepcke. Note sur l’ expression (((aa)a)... )a, et les fonctions inverses correspon-

dantes. Journal Fur Die Reine und Angewandte Mathematik [=Crelle’s Journal],

42:83–90, 1851.

[197] V. Wood. Quaternions. The Analyst, 1(7):11–13, Jan. 1980.

[198] E.M. Wright. On a sequence deﬁned by a non-linear recurrence formula. Journal

of the London Mathematical Society, 20(1):68–73, 1945.

[199] E.M. Wright. Iteration of the exponential function. Quarterly Journal of Mathe-

matics (Oxford), 18:228–235, 1947.

[200] E.M. Wright. A class of representing functions. Journal of the London Mathematical

Society, 29(1):63–71, 1954.

[201] A. P. Yefremov. Quaternions: Algebra, geometry and physical theories. Hypercom-

plex Numbers in Geometry and Physics, 1:104–119, 2004.

[202] S. Yeshurun. Reverse order exponentiation. School Science and Mathematics,

89(2):136–143, Feb. 1989.

[203] G. Zirkel. But does it converge? Mathematics and Computer Education, 18(2):153,

Spring 1984.

[204] A.E. Dubinov and I.N. Galidakis. An explicit solution to Kepler’s equation. Physics

of Particles and Nuclei. Letters, 4(3):213–216, May 2007.

[205] I.N. Galidakis. On the enumeration of the roots of arbitrary separable equations

using hyper-Lambert maps. Research and Communications in Mathematics and

Mathematical Sciences, 12(1):1–15, 2020.

[206] I.N. Galidakis. Lambert’s W function and convergence of inﬁnite exponentials in

the space of quaternions (corrigendum). Complex Variables, 52(4):351, Apr. 2007.

[207] I.N. Galidakis. Lambert’s W function and convergence of inﬁnite exponentials in

the space of quaternions. Complex Variables, 51(12):1129–1152, Dec. 2006.

[208] I.N. Galidakis. On some applications of the generalized hyper-Lambert functions.

Complex Variables and Elliptic Equations, 52(12):1101–1119, Dec. 2007.

[209] I.N. Galidakis. On solving the p-th complex auxiliary equation f(p)(z) = z.Complex

Variables, 50(13):977–997, Oct. 2005.

[210] I.N. Galidakis. On an application of Lambert’s W function to inﬁnite exponentials.

Complex Variables Theory Appl., 49(11):759–780, Sep. 2004.

[211] I.N. Galidakis and E.W. Weisstein. Power tower, from mathworld–a wolfram web

resource., 2011. http://mathworld.wolfram.com/PowerTower.html.

45