A Collection of References For Infinite Exponentials and Tetration
Most of the author's research on tetration and infinite exponentials was/is possible because Dave L. Renfro has kindly provided him over a course of 5 years with what is probably the most extensive list of references relating to these subjects. Not all of them are immediately relevant and where appropriate, the author gives a quick summary of their contents (for even more references, consult Paper 1). All the references are sorted alphabetically on the first author or problem proposer's name.
Problems, Notes, 7- 9, American Mathematical Monthly, 28(3) (March 1921), 140 - 143. A discussion on the equation x^{y}
= y^{x} along with some short notes on its parametrization.
Problems for Solution, 5101-5110. The American Mathematical
Monthly, 70(5) (May 1963), 571 - 573. A collection of
problems with one relevant to the equation m^{n} = n^{m}.
Jeffrey M. Alden , A Closer Look at x^{xx...} . Journal of Undergraduate Mathematics,
11(2) (Sep. 1979), 63-66. Parametrization of the locus of the period 2 iterated exponential.
Daniel S. Alexander, A History of Complex Dynamics, From Schroeder
to Fatou and Julia. [Reviewed by R. B. Burckel, SIAM Review, 36(4)
(December 1991)]. ISBN 3-528-06520-6. Shroeder's fixed Point
Theorem and a good discussion from Schroeder, Korkine and Faraks on the
existence of a function that could possibly extend a function's iterates to
non-integral or even complex values.
Arnold O. Allen, e^{π} or π^{e}?. Journal of Recreational Mathematics, 2(4), (October 1969), 255 - 256. Elementary analysis of when the different exponentials are the same.
M.M. Alliaume, Sure Le Developpment En Serie De La Racine D' Une Equation. Mathesis Requeil Mathematique, 44, (1930), 163 - 166. Series development for a solution to Kepler's equation.
Joel Anderson, Iterated Exponentials, The Mathematical Association of America,
111 Monthly (October 2004), 668 - 679. A short presentation on the sequence a, a^{a}, a^{aa}.
J. C. Appleby, Notes on Hexponentiation, The Mathematical Gazette,
79(484) (March 1995). A short investigation on finding
expressions for h(x), such that h(h(x)) = exp(x).
Fashid Arjomandi, Problem 27.1, Mathematical Spectrum, 27(3) (1994/1995), 68. Number theoretic tetration problem about divisibility.
R.C. Archibald, R. Clare, Problem 884, Mathematics of Computation, 5(35), (July 1951), 140. Solutions to the first and second auxiliary tetration equations.
R.C. Archibald, R. Clare, Problem 136, Mathematics of Computation, 6(39) (July 1952), 204. Solution of y=e^{y}.
J. Marshall Ash, The Limit of x^x^...^x as x Tends to Zero,
Mathematics Magazine69 (1996), 207 - 209. Short paper investigating the
behavior of the hypertower when x approaches 0. Interesting connections with the author's
notation ^{+∞}x in article 3.
H.D' Assumpcao, G. Grossley, R.J. Armstrong, Note 2891. Iterated Exponentials,
The Mathematical Gazette44 #348(May 1960), 126-127.
F. Azarpanah, Proof Without Words: Convergence of a Hyperpower Sequence,
Mathematics Magazine77 (2004), 393.
I. N. Baker, P. J. Rippon, Convergence of Infinite Exponentials. Annales Academiae Scientiarium Fennicae, Series A.I. Mathematica, 8, (1983), 179 - 186. One of the most fundamental papers on infinite exponentiation. The main result of the paper completely characterizes the main cardioid or nephroid in the complex plane where convergence occurs. A must read. Domains of convergence on the complex plane displayed similar to the fractal on article 1,
I. N. Baker, P. J. Rippon, A Note on Complex Iteration. American
Mathematical Monthly, 92(7), (Aug. - Sep. 1985), 501 - 504.
Perhaps THE definitive reference for the convergence of complex
iterates of f(z) = c^{z}, along with some very nice diagrams that show
the domains of convergence on the complex plane similar to the fractal on article 1.
I. N. Baker, P. J. Rippon, Towers of Exponents and Other Composite Maps. Complex Variables,
12, (1989), 181 - 200. A followup of the previous reference with detailed discussions on the area of convergence of not only the iterated exponential, but other contractive maps as well, with additional preliminary results about the two-cycle convergence around the origin and graphical representations of some basins of attraction.
I. N. Baker, P. J. Rippon, Iterating Exponential Functions With Cyclic Exponents. Mathematical Proceedings of the Cambridge Philosophical Society,
105, (1989), 359 - 375. An extensive analytical treatise of convergence of exponential towers with the exponents being a cyclic sequence, {a1, a2, ..., an, a1,...}.
D. F. Barrow, Infinite Exponentials. American Mathematical Monthly,
43(3) (March 1936), 150 - 160. A very rigorous treatise
of the infinite exponentials cases with real exponents, with lots of
theorems and detailed graphs on the behavior of the iterates. Contains lemmas
for some cases where the iterative bases are not all the same, using x^{1/x}
decompositions of the iterate bases, similar to the author's analysis on article
1.
Morris H. Barshinger , A Look at x^{xx...} . Journal of Undergraduate Mathematics,
5(1) (March 1972), 37-41.
E. T. Bell, The Iterated Exponential Integers. The Annals of
Mathematics, Second Series, 39(3), (July 1938), 539 - 557. An
extensive, quite involved and rigorous treatise of iterated exponential integers,
with connections to Stirling Numbers, Catalan Numbers, Bernoulli
Polynomials Combinatorial identities and number Theory.
E. T. Bell, Iterated Exponential Numbers. Bulletin of the American Mathematical Society, 43, (1937), 774 - 775. A quick note on some number theoretic properties of a certain class of exponentially defined integers.
W. W. Beman, Problem 389. American Mathematical Monthly, 21(1) (January 1914), 23. A series expansion of e^{ex}.
Carl M. Bender, Advanced Mathematical Methods for
Scientists and Engineers, Approximate Solution of Nonlinear Differential Equations,
ISBN 0-07-004452-X. A solution to y = exp(xy).
Bob Bertuello, An Infinite Exponential. Mathematical Spectrum, 27(1) (1994/95), 22. One more reader discovers the fascinating convergence of the iterated exponential.
Louis Brand, Binomial Expansions in Factorial Powers. American Mathematical Monthly, 67(10) (December 1960), 953 - 957. An interesting definition of "power" using factorials, which leads to several interesting binomial identities.
A. Dubinov and I.N. Galidakis, Explicit Solution of the Kepler Equation. Physics of Particles and Nuclei, Letters, 4(3) (May 2007), 213 - 216. Using a certain family of Lambert W-like functions to solve Kepler's equation exactly.
Henk J.M. Bos, Johann Bernoulli on Exponential Curves, ca. 1695 Innovation and Habituation in the Transition from Explicit Constructions to Implicit Functions. Nieuw Archief voor WisKunde, 14(4) (1996), 1 - 19. Historical notes on Bernoulli and Leibnitz on issues of exponential Calculus, with section 7 investigating the curve y=x^{x}.
J. G. Hagen, S. J., On the history of the extensions of the Calculus. Bulletin of the American Mathematical Society, 6 (1899-00), 381 - 390. Section IV contains a detailed discussion on the differentiability of the iterates of the exp function, exp_{n}(x).
Harlan J. Brothers and John A. Knox, New Closed-Form Approximations to the Logarithmic Constant e. Mathematical Intelligencer, 20(4) (Fall 1998), 25 - 29. A very nice exposition of various approximations of the base of Naperian logarithms with a short note on hyper-exponentiation at the end. Recommended reading.
N.D. Hayes, The Roots of The Equation x=(c*exp)^{n}x And The Cycles of The Substitution (x|ce^{x}). Quarterly Journal of Mathematics (Oxford), 3(2) (1952), 81 - 90. Very nice exposition for the roots of the equation x=(c*exp)^{n}x, where (c*exp)^{n} denotes the iterated substitution sequence x|->c*exp(x), with conditions for existence for periodic solutions in the complex plane. Recommended reading.
Florian Cajori, Napier's Logarithmic Concept. American Mathematical Monthly, 23(3) (March 1916), 71 - 72. Various issues related to Napier's use of the logarithm and computations.
Florian Cajori, History of the Exponential and Logarithmic Concepts. American Mathematical Monthly, 20(1) (January 1913), 5 - 14. Fascinating historical journey on the origin of various exponential and logarithmic concepts from Euler to the 20-th century. Recommended reading.
Florian Cajori, History of the Exponential and Logarithmic Concepts (part 2). American Mathematical Monthly, 20(2), (February 1913), 35 - 47. Part 2 of above.
Errata in the February Issue. American Mathematical Monthly, 20(3) (March 1913), 104. Errata of the above.
Florian Cajori, History of the Exponential and Logarithmic Concepts (part 3). American Mathematical Monthly, 20(3) (March 1913), 75 - 84. Part 3 of above.
Florian Cajori, History of the Exponential and Logarithmic Concepts (part 4). American Mathematical Monthly, 20(4) (April 1913), 107 - 117. Part 4 of above.
Florian Cajori, History of the Exponential and Logarithmic Concepts (part 5). American Mathematical Monthly, 20(5) (May 1913), 148 - 151. Part 5 of above.
Florian Cajori, History of the Exponential and Logarithmic Concepts (part 6). American Mathematical Monthly, 20(6) (June 1913), 173 - 182. Part 6 of above.
Florian Cajori, History of the Exponential and Logarithmic Concepts (part 7). American Mathematical Monthly, 20(7) (September 1913), 205 - 210. Part 7 of above.
H. S. Carslaw, Relating to Napier's Logarithmic Concept. American Mathematical Monthly, 23(8) (October 1916), 310 - 315. Comments on the Cajori article and a response by professor Cajori.
R. D. Carmichael, Problem 275. American Mathematical Monthly, 14(2) (February 1907), 27. Solution to equations x^{y}-y^{x}=0 and y-x=a*(a+1)^{1/a}.
Arthur Latham Baker, Functional Exponents. School Science and Mathematics, 8(3) (March 1908), 225 - 227. Elementary discussion on what f^{a}(x) means for integer and fractional a.
E. Barbette, Des Progressions Logarithmiques. Mathesis Requeil Mathematique, 8(2) (1898), 135 - 137. Sequences whose exponents reduce to geometric progressions.
Albert A. Bennett, The Iteration of Functions of One Variable. The
Annals of Mathematics, Second Series, 17(1) (September 1915),
23 - 60. General and quite elaborate discussion on iterating
real and complex functions of one variable, including series and possible
representations of the iterations as matrix multiplication.
Albert A. Bennett, Note on an Operation of the Third Grade.
The Annals of Mathematics, Second Series, 17(2) (December 1915), 74
- 75. Short excerpt on the analytic continuation of iterated exponentiation.
Nick Bromer, Superexponentiation, Mathematics Magazine, 60
(1987), 169 - 174. Very nice paper on the two hyper4 operators, (both
the "upper" hyper4 and the "lower" hyper4 one) and on hyperexponential functions
formed via their combinations with short notes on a possible continuation
for rational powers using limits.
Michael Brozinsky, Problem 4394. School Science and Mathematics, 93(6) (October 1993), 340. A finite tetration problem and a related inequality.
R. A. Bruce (student), General repeated exponentiation. The American Mathematical Monthly, 71(8) (October 1964), 759 - 761. Bounds of convergence for iterated exponential.
Barry W. Brunson, The Partial Order of Iterated Exponentials.
The American Mathematical Monthly, 93(10) (December 1986), 779
- 786. An excellent and in depth discussion that puts to rest
the question of the implicit order of iterated exponentials and in particular
questions such as, π^e^π^e <?> e^π^e^π. Interesting connections with
Group Theory.
E. P. Bugdanoff, Problem 3515 [1931,462]. American Mathematical Monthly, 39(9) (November 1932), 552 - 555. Elementary analysis of the equation 2^{x}=4*x.
L. E. Bush, The 1961 William Lowell Putnam Mathematical Competition.
The American Mathematical Monthly, 69(8) (October 1962), 759
- 767. One problem relating to the equation x^{y} = y^{x}.
F. A. Butter, A note on the equation x^{y}=y^{x}. American Mathematical Monthly, 46(6), (June/July 1939), 316 - 317. Short note on the real solutions of the aforementioned equation.
W. D. Cairns, Problem 422. The American Mathematical Monthly, 22(4) (April 1915), 133. The solution to the equation x^{x*sqrt(x)}=(x*sqrt(x))^{x}.
R. D. Carmichael, On a Certain Class of Curves Given by Transcendental
Equations. The American Mathematical Monthly, 13(12) (December
1906), 221 - 226. Perhaps THE definitive analysis for the family
of curves x^{y} - y^{x}.
R. D. Carmichael, On Certain Transcendental Functions Defined
by a Symbolic Equation. The American Mathematical Monthly, 15(4) (April 1908), 78 - 83. A generalization of the previous discussion.
R. Cooper, The Relative Growth of Some Rapidly Increasing Sequences. Journal of the London Mathematical Society, 29(1) (1954), 59-62. The general rate of growth of sequences defined by the recursion a_{n+1}=a^{an}.
N.A. Court, Solution to Monthly Problem 3649. American Mathematical Monthly, 56 (1949), 414-415. Solving the functional f(xy)=[]f(x)^{yb}[f(y)^{xa}].
Peter Colwell, Bessel Functions and Kepler's Equation. American Mathematical Monthly, 99(1) (January 1992), 45 - 48. Lagrange's and Bessel's solutions to the famous Kepler equation.
Robert M. Corless, David J. Jeffrey, and Donald E. Knuth,
A Sequence of Series for Lambert's W function. Department of Applied Mathematics,
University of Western Ontario, London, CANADA, N6A 5B7. A uniform treatment
of several series expansions for the Lambert's W function, leading to an infinite
family of new series. The standard paper on the most basic properties
of the Lambert's W function, with some connections to infinite processes and
in particular infinite exponentials. A must read.
Charles C. Cross, Problem 93. The American Mathematical
Monthly, 6(8/9) (August/September 1899), 199-201. System of two equations of the form x^{y}+/-y^{x}=c.
H. Martyn Cundy, x^y=y^x: An investigation. The Mathematical Gazette, 71(456) (June 1987), 131-135.
Philip J. Davis, Leohard Euler's Integral: A Historical Profile of the Gamma Function. The American Mathematical
Monthly, 66(10) (December 1959), 849 - 869. Historical development of the Gamma function, with various attempts to extend the factorial, such as that of Hadamard. Connections with the zeta function and the incomplete beta functions. Descriptions of the Gamma function on the complex plane. Recommended reading.
Ed. Delville, Nombres Egaux a Leurs Logarithmes. Mathesis, 4(2) (1912), 264. Discussion on when a number is equal to its logarithm.
Edward Drake Roe, Jr. A. M., Problem 68. The American Mathematical
Monthly, 5(4) (April 1898), 110. Tetration problem for fixed lower base and variable exponents.
Daniel Drew (student), A recursion relation involving exponentials. The American Mathematical
Monthly, 56(9) (November 1949), 660 - 664. General solution for the recursion of iterated exponentials.
Richard M. Dudley, Problem 5098 [1963,445]. The American Mathematical Monthly, 71(5) (May 1964), 563. Existence of non-constant entire f such that f(z)=exp_{n}(f_{n}(z)), with f_{n} being the iterates of exp.
G. Eisenstein, Entwicklund von a^a^a^.... Journal Fur Die Reine Und Angewandte Mathematik, 28(8) (1944), 49 - 52. Various results concerning the infinite exponential and in particular investigations of the solutions of the equation x^x^n = y, as well as the corresponding series expansions.
G. Enestrom, Es Ware Ratsam, Den Passus, Bibliotheca Mathematica, 13(3) (1912 - 1913), 270. Short note on parametrization (originally by Euler) of the locus of the equation x^{y}=y^{x}.
Carlton Lee Evans, Interesting Logs, Reader's Reflection Column, Mathematics Teacher, 72(7) (October 1979), 489. Elementary analysis of the function x^{1/x}.
C. J. Everett, Jr., An Exponential Diophantine Equation, American Mathematical Monthly, 43(4), (April 1936), 229 - 230. Rational solutions of the equation x^{y}=y^{x}.
Comment 8, Bulletin of the American Mathematical Society, 12 (1905-06), 115. [8-th regular meeting of the San Franscisco Section of the AMS, University of California, 30 September 1905]. Short note on higher arithmetical operations.
M. Jules Farkas, Sur Le Fonctions Iteratives. Journal de Mathematiques Pures et Appliquees [=Liouville's Journal], 10(3) (1884), 101 - 108. Analytic iteration of functions and Schroeder's equation.
Steven R. Finch, Iterated Exponential Constants. Encyclopedia of Mathematics & its Applications, Cambridge University Press, 2003, 448 - 452. Analytic iteration of functions and Schroeder's equation.
Philip Franklin, Discussions, Relating to the Real Locus Defined
by the Equation x^{y} = y^{x}. The American Mathematical
Society, 24(3) (March 1917), 137. Discussion similar
to the above, but not as elaborate.
B. Frizell, The problem of defining the set of real numbers. Bulletin of the American Mathematical
Society, 17 (1910-11), 296. [28-th regular meeting of the Chicago section of the AMS, University of Chicago, 28-30 December 1910]. Transfinite series of fundamental sequences and arithmetical operations. Connections with Ackermann's function.
Daniel Geisler, Tetration.org Page. Lots of work on the dynamics of the map z|->c^{z}, with fractal plots and subplots, orbit trajectories and a proposed solution for extending Tetration to the real numbers, which for some strange reason is not widely known. The author appears likely to be the first who has identified the full extent of the basin of period 2 on the iteration of this map on the Complex plane using FractInt and was probably among the first ones who realized that there exist attracting basins of period p, for any p > 1. Numerical work on the iterates of the map z|->c^{z} by this author has been used to validate and verify several later important theoretical results and Theorems about the dynamics of this map in subsequent articles.
I.N. Galidakis, On An Application of Lambert's W Function to Infinite
Exponentials. Complex Variables, 49(11) (September 2004), pages 759-780.
Quite a long paper, recently reviewed in which the author deals with more elegant
ways to prove convergence for iterated exponentials for the real and
complex cases and where applications to fractals and explicit series
expansions for all exponentials z^^n are given.
I.N. Galidakis, On Solving the p-th Complex Auxiliary Equation f^{(p)}(z)=z.
Complex Variables, 50(13) (October 2005), pages 977-997. An extension of the results of the
previous paper, which exhibits expressions for finding fixed points of
period p>=1 of the iterated exponential z^z^...^z and classifies the
periodic domains of the iterates of f(z)=c^{z}.
I.N. Galidakis, Lambert's W Function and Convergence of Infinite Exponentials in the Space of Quaternions. Complex Variables, 51(12) (December 2006), pages 1129-1152. An extension of Lambert's W function to the space of Quaternions, which is used to generalize the results about the convergence of the iterated exponential z^z^...^z of the previous papers in the space of Quaternions, with a graphical application of the results which produces 3D representations of the associated Mandelbrot and Julia sets, using a quaternion fractal program.
I.N. Galidakis, On Some Applications of The Generalized Hyper-Lambert Functions. Complex Variables, 52(12) (December 2007), pages 1101-1119. A study of the properties of a new class of functions which generalize Lambert's W function in the direction of higher order exponentials, with applications to algebra, differential calculus, tetration and Kepler's equation.
Raymond Garver, On the Approximate Solution of Certain Equations. American Mathematical Monthly, 39(8) (October 1932), 476 - 478. A discussion of the solutions of the equation x^{x}=c.
F. Gerrish, a^b=b^a: the positive integer solution. The Mathematical Gazette, 76(#477) (Nov. 1992), 403.
Ghunaym M. Ghunaym, Problem 4696. School Science and Mathematics, 99(1) (January 1999), 54. Infinite tetration of sqrt(2) with integer bases m and n.
Richard A. Gibbs, Problem 4334. School Science and Mathematics, 92(5) (May/June 1992), 292-293. Intersection of the graphs of a^{x} and log_{a}(x).
Jekuthiel Ginsburg, Iterated Exponentials. Scripta Mathematica II, (1945), 340 - 353. Relationships on the coefficients of the series of the iterated exponential function exp_{n}(x).
J.W.L. Glaisher, A Theorem In Trigonometry. Quarterly Journal of Pure and Applied Mathematics, 15 (1878), 151 - 157. Trigonometric relationships of the roots of Kepler's equation.
Peter Goodstein, Note 2803 Limits of iterated logarithmic functions. The Mathematical Gazette, 42(#342) (Dec. 1958), 295-296.
R.L. Goodstein, Note 2804. The system of equations b^{x}=x_{1}, b^{x1}=x_{2},..., b^{xn}=x. The Mathematical Gazette, 42(#342) (Dec. 1958), 296-299.
R.L. Goodstein, The Equation a^{b}=b^{a}. The Mathematical Gazette, 28(279) (May 1944), 76. Quick and elegant analysis of both integral and non-integral solutions of the infamous equation.
Ralph Greenberg, Problem E 1597 [1963, 757]. American Mathematical Monthly, 71(3) (March 1964), 322. Maximization of iterated exponentials related to permutation of the bases.
M. J.-H. Grillet, Les Exponentielles Successives D'Euler. Journal de Mathematiques Pures et Appliquees [=Liousville's Journal], 10 (1985), 233 - 241. Another extensive treatise of the convergence of the infinite exponential, with deep results into the solution of the auxiliary equations for the period 2 real region.
A. S. Hendler, Problem E 1144 [1954,711]. The American Mathematical
Monthly, 62(6) (June/July 1955), 446. For what positive values of a is log_{a}(b)< b for all positive b?
V. F. Ivanoff, Problem E 34 [1933,241]. The American Mathematical Monthly,
44(2) (February 1934), 104 - 106. Solutions of the equation x^{-x}=(-x)^{x}.
M. Kossler, On The Zeros of Analytic Functions. Proceedings of the London Mathematical Society,
19(2) (1921), back pages for the January 13, 1921 Meeting. A very nice classification of the roots of certain types of algebraic combinations of analytic functions (very similar to the ideas of the Hyper-Lambert functions!).
F. Gobel, R. P. Nederpelt, The Number of Numerical Outcomes
of Iterated Powers. The American Mathematical Monthly, 78(10)
(December 1971), 1097 - 1103. An in depth combinatorial analysis for the
possible outcomes of exponential towers under different groupings of
the exponential itself. Connections with Catalan numbers and calculation of
upper bounds.
David Hammer, Problem 171, Parenthetical Roots, Mathematics and Computer Education, 17(1) (Winter 1983), 73. Finite tetration problem for sqrt(2).
Alvin Hausner, Problem E 1474 [1961,573], The Equation m^{nm} = n^{mn}. The American Mathematical Monthly,
69(2) (February 1962), 169. Shows that the aforementioned equation has no solution in integers m <> n.
Alvin Hausner, Algebraic Number Fields and the Diophantine
Equation m^{n} = n^{m}. The American Mathematical Monthly,
68(9) (November 1961), 856 - 861. An in depth group-theoretic
analysis, searching for solutions of the above equation in algebraic
number fields, including the complete solution for the case of integers.
Jean Van Heijenoort, From Frege to Goedel: A Source Book in Mathematical Logic, 1879 - 1931, Harvard University Press, 1967, 493 - 507. Section "On Hilbert's construction of the real numbers", written by Wilhelm Ackermann (1928). Extensive analysis of the Ackermann function and tangential issues related to its applications in logic and real analysis.
J. L. Hickman, Analysis of an Exponential Equation with
Ordinal Variables. Proceedings of the American Mathematical Society, 61(1)
(November 1976), 105 - 111. An extensive discussion on the solutions z of
x^{y} = y^{z}, with x and y transfinite ordinals.
J. Higgins, On Note 3232, The Mathematical Gazette 55#391 (Feb. 1971), 63-64.
Solomon Hurwitz, On the Rational Solutions of m^{n}=
n^{m}, with m<>n. The American Mathematical Society, 74(3)
(March 1967), 298 - 300. Discussion on the complete parametrization of
all rational solutions to the equation m^{n} = n^{m}.
R. Arthur Knoebel, Exponentials Reiterated. The American
Mathematical Monthly, 88(4) (April 1981), 235 - 252. Perhaps
the classic article on iterated exponentiation. Summary of most problems
related to infinite exponentiation, connections with biology, short comments on
the analytic continuation of the hyper4 operator and the ackermann function,
with many references, many of which are included here.
Nishizawa, Kiyoko, Convergence of a^a^a^... and bifurcation of dynamical systems. Sci. Rep. Res. Inst. Eng., 11(1988), 54-60.
Y. S. Kupitz, H Martini, On The Equation x^{y} = y^{x},
Elemente der Mathematik, 55 (2000) 95 -101. A quite thorough investigation
of the above equation with a parametrization of its non-trivial solutions.
Honsta. W. Labbers, Jr., Problem 420, Nieuw Archief voor WisKunde, 3(24) (1976), 207-210. A number-theoretical problem related to the Ackermann function.
Paul Heckbert (student), |x|^{y}=|y|^{x}, The Mathematics Student Journal, 22(4) (April 1975), 4,7. Quick analysis of the equation |x|^{y}=|y|^{x}.
L. J. Lander, Problem E 1124 [1954,423],
The American Mathematical Monthly, 62(2) (February 1955), 124 - 125. Solutions of the equation m^{n}=n^{m}, for integer m and n.
Helmut Langer, An Elementary Proof of the Convergence of Iterated
Exponentials, Elemente Math., 51 (1996), 75-77. A very nice exposition
of the basic properties of the iterative process x->a^{x}.
J. Lense, 3053. The American Mathematical Monthly, 31(10) (December 1924), 500 - 501. A short problem on the real
infinite exponential iteration.
H.W. Lenstra, Jr., 3053. Problem 566, Nieuw Archief voor WisKunde, 3(28) (1980), 300-302. Short problem on the growth of the function 2^{2n}.
J.E. Littlewood (ref), Note 2285, A Note on Multiple Exponentials, The Mathematical Gazette 36#316 (May 1952), 127 - 128.
J.E. Littlewood (ref), Note 2517, On Note 2285, The Mathematical Gazette 39#328 (May 1955), 141 - 142.
Sidney Luxton Loney, Plane Trigonometry, 5'th Edition, Cambridge University Press, parts I and II, (1925), 203. Short note on the trigonometric form of x^x^x^....
George F. Lowerre, A Logarithm Problem and How It Grew, The
Mathematics Teacher, 72 (1979), 227 - 229. A nice short investigation on
the solutions of the equation a^{x} = x, which bares interesting
connections with the function f(x) = x^{1/x} as it was investigated in article 1.
J. Van De Lune, Problem 899, Nieuw Archief voor Wiskunde, 4(13) (1995), 251 - 252. Problem related to iterated exponentiation sequence.
J. Macdonnell, Some Critical Points on the Hyperpower
Function ^{n}x = x^x^..., Int. J. Math. Educ. Sci. Technol., 20(2), 1989,
297-305. An in depth analysis of the critical points of the real HyperPower
function, both for finite n and for n->+∞, along with some detailed graphs on
what happens at 0^{+} and on the interval, (0,(1/e)^{e}).
A. J. Macintyre, Convergence of i^i^i^..... Proceedings
of the American Mathematical Society, 17(1) (February 1966), 67. A
very short expose on why the partial iterates i, i^i, i^i^i, ... eventually
converge. With a short graphic for the attractor. (Compare the attractor
with the one shown in article 1 for the Julia
set for i^i^i^....).
Warren B. Manhard 2d, Is Exponentiation Commutative? Mathematics Teacher, 74(1) (January 1981), 56 - 60. Exploration of the equation a^{b}=b^{a}, with graphical plots of the solution locus for some reals.
Robert L. Mayes, Discovering Relationships, Logarithmic and Exponential Functions. School Science and Mathematics, 94(7) (November 1994), 367 - 370. Exploration and comparison of the families of functions f(x)=b^{x} and g(x)=log_{b}(x).
Patrick J. McCarthy, Decompositions of Functions, Functional nth roots of unity (in functional equations). Mathematical Gazette, 64(428) (June 1980), 107-115.
J.S. Griffith, Decompositions of Functions, Some approaches to the general problem (in functional equations). Mathematical Gazette, 64(428) (June 1980), 116-120.
M. C. Mitchelmore, A Matter of Definition. The American
Mathematical Monthly, 81(6) (June - July 1974), 643 - 647. A
good discussion that arose from the question of whether the equation x^x^...=2
has a solution, along with an investigation of the graph of y(x) = x^{1/x}
and the associated functions for some cases on the iterated
exponentiation.
E. J. Moulton, The Real Function Defined by x^{y} = y^{x}.
The American Mathematical Monthly, 23(7) (September 1916), 233 - 237. Another detailed discussion concerning the real loci of the above equation, one that Knoebel uses heavily in his article.
Joseph Morton, Graphical Study of x^{x}=y.
The Mathematical Student Journal, 9(2) (January 1962), 5. Analysis of the equation x^{x}=y using complex numbers.
R. P. Nederpelt, E1903. The American Mathematical
Monthly, 79(4) (April 1972), 395 - 396. A problem on how many
different integers one obtains using different bracketing orders in the
exponentiation of a tower of iterated 2's.
D. J. Newman, Problem 4569 [1954,51],
The American Mathematical Monthly, 62(3) (March 1955), 190 - 191. Does there exist a function f such that f(f(x)) ~ e^{x} (in terms of order of magnitude)?.
Sam Newman (proposer), Problems and Solutions , Mathematics Magazine, 40(1967), 283-284. Exponential Derivative.
Ivan Niven, Which is Larger, e^{π} or π^{e}?,
Two-Year College Mathematics Journal, 3(2) (1972), 13 - 15. [Reprinted on pp. 445
- 448 of Tom M. Apostol, et al (editors) A Century Of Calculus, Part II,
1969 - 1971, MAA, 1972]. Comparison of the two exponentials.
C. S. Ogilvy, E853. The American Mathematical Monthly,
56(8) (October 1949), 555 - 556. The shortest solution to the
infinite exponentiation problem, very similar to the author's approach in article 1.
Jean J, Petersen and Peter Hilton, exp(π) > π^{e}? Readers Reflections Column, Mathematics Teacher, 74(7) (October 1981), 501 - 502. A discussion of the "Is Exponentiation Commutative" article on the same magazine.
Hillel Poritsky, Problem 2851 [1920,377]. The American Mathematical Monthly,
29(3) (March 1922), 132 - 133. Short discussion on the existence of an analytic function satisfying f(x+1)=e^{f(x)}.
Louis O'Shaughnessy, Problem 433. American Mathematical Monthly, 26(1) (January 1919), 37-39. Fractional derivative problem.
Stanley Rabinowitz, Problem 191, Exponential Roots, Mathematics and Computer Education, 18(2) (Spring 1984), 150-151. Solutions of the equation x^{x}=c for various c.
W. R. Ransom, Problem E 3 [1932,489]. American Mathematical Monthly, 40(2) (February 1933), 113. The equation y=x^{x} again.
Dave L. Renfro, Exponential and Logarithmic Commutativity,
The Mathematics Teacher, 91 (1998), 275, 362. A nice discussion on
the analytic solvability of equations, exp(10^{x}) = 10^{exp(x)},
ln(log(x)) = log(ln(x)) and extending the ideas to the more general
classes of equations, a^{b^x} = b^{a^x} and log_{a}(log_{b}(x))
= log_{b}(log_{a}(x)).
John Riordan, A Note on Catalan Parentheses. The American Mathematical
Monthly, 80(8) (October 1973), 904 - 906. The problem
of finding the distinct products formed by inserting appropriate parentheses
in between the numbers. Catalan numbers.
R. Robinson Rowe, The Mutuabola. Journal of Recreational Mathematics, 3(3) (July 1970), 176 - 178. Analysis of the solutions of the equation x^{y}=y^{x}.
Bertram Ross and Baldev K. Sacheva, The Inversion of Kepler's Equation. Int. J. Math. Educ. Sci. Technol., 16 (1985), 558 - 560. Solution to Kepler's equation by inverting its series expansion.
Daihachiro Sato, Shorter Notes, Algebraic Solution of x^{y}
= y^{x} (0<x<y). Proceedings of the American Mathematical Society, 31(1) (January 1972), 316. The characterization of pairs of algebraic integers which are commutative with respect to exponentiation.
J. Scheffer, Problem 307. The American Mathematical Monthly, 16(2) (February 1909), 32 Find x and y if y^{x}=2 and x^{y}=3.
E. D. Schell, Problem E 640 [1944,472]. American Mathematical Monthly, 52(5), (May 1945), 278 - 279. Short note on integer solutions of the equation x^{y}=y^{x}.
Donald L. Shell, Convergence of Infinite
Exponentials. Ph.D. Thesis, University of Cincinnati (1959). Shell's original Ph.D. thesis on the convergence of Infinite Exponentials, with some deep results and theorems. Required reading for anyone doing serious work on tetration.
Donald L. Shell, On the Convergence of Infinite
Exponentials. Proceedings of the American Mathematical Society, 13(5) (October
1962), 678 - 681. Another excellent paper on the convergence of the
complex infinite exponentials and the establishment of some of the complex
regions where convergence occurs.
Thomas W. Shilgalis, Graphical Solution of The Equation a^{b}
= b^{a}. The Mathematics Teacher, 66(3) (Mar. 1973), 235. Short
paper which arose from the question on whether e^{π} <?>
π^{e}.
Steven L. Siegel, Exponential Equations. Reader Reflections Columnm, Mathematics Teacher, 100(4) (November 2006), 238. Solutions of the equation x^{a}=a^{x}.
Wactaw Sierpinski, (pp. 106-109 in) Elementary Theory of Numbers. Monografie Matematyczne, 42 (1964), 106-109.
David J. Silverman, What's the Limit? Journal of Recreational Mathematics, 4(2) (April 1971), 144. Is the limit of x^{x...} equal to 2 or 4?
Man-Keung Siu, An Interesting Exponential Equation Mathematical Gazette, 60(413) (Oct. 1976), 213-215. The equation: x1^x2=x2^x3=...=xn^x1.
H. L. Slobin, The solutions of x^{y} = y^{x}, x>0,
y>0, x<>y and Their Graphical Representation. The American Mathematical
Monthly, 38(8) (October 1931), 444 - 447. The
rational loci of the above equation along with nice parametrizations for x and y.
G. P. Speck, a^{b} Versus b^{a}. School Science and Mathematics, 65(6) (June 1965), 489 - 490. The two different powers compared.
V. M. Spunar, Problem 430. American Mathematical Monthly, 26(9) (November 1919), 415. Solution of the equations x^{y}+y^{x}=x*y and x^{y}+y^{x}=x+y.
R. M. Sternheimer, On Certain Integers Which Are Obtained
By Repeated Exponentiation. Journal of Recreational Mathematics, 22(4), 271 - 276, 1990. A short exposition on calculating towers of integer
exponentials.
L. John Stroud, On x=a^{x}. The Mathematical Gazette, 53(385) (Oct. 1969), 289-293.
Karl R. Stromberg. (pp. 185-186 of) Introduction to Classical Real Analysis, Wadswath 1981, 185-186.
W. R. Thomas, John Napier. The Mathematical Gazette, 19(234) (June 1935), 192-205. A very nice historical summary on the origins of the word "logarithm", tracing back to Archimedes.
W. J. Thron, Convergence of Infinite Exponentials with
Complex Elements. Proceedings of the American Mathematical Society, 8(6)
(December 1957), 1040 - 1043. A slightly more general and elaborate
discussion on when the infinite complex exponential converges, with conditions
given for the iterative bases and the limit whenever convergence occurs.
W. J. Thron, Convergence Regions for Continued Fractions
and Other Infinite Processes. The American Mathematical Monthly, 68 (October
1961), 734 - 750. A general discussion of some iterative processes,
including continued fractions, exponential towers, etc, with several
interesting theorems on their regions of convergence (establishment of the Shell-Thron region of convergence for infinite exponentials, found also in article
1).
Smith D. Turner, Under What Conditions Can a Number Equal its Logarithm?. School Science and Mathematics, 27(7) (October 1927), 750-751. Connection with the function y=x^{1/x}, and subsequent analysis.
Smith D. Turner, Under What Conditions Can a Number Equal its Logarithm? (part 2). School Science and Mathematics, 28(240) (April 1928), 376 - 379. Continuation of the discussion of the previous.
H. S. Uhler, On the Numerical Value of i^{i}. American Mathematical Monthly, 28(3), (March 1921), 114 - 121. Numerical estimates for various fundamental exponential constants, followed by historical notes on the relation i^{i}=exp(-π/2) by R. C. Archibald.
John T. Varner III, Comparing a^{b} and b^{a} Using Elementary Calculus.
Two-Year College Mathematics Journal (now College Mathematics Journal),
7(4) (December 1976), 46. Short paper on determination of a^{b}
<?> b^{a}, with a corollary for e and π.
Glenn T. Vickers, Experiments With Infinite Exponents. Mathematical Spectrum, 27(2) (1994/95), 34. Discussion on "An Infinite Exponential" on the same magazine.
Glenn T. Vickers, More About An Infinite Exponential. Mathematical Spectrum, 27(3) (1994/95), 54 - 56. Elementary analysis of some iterated exponential sequences.
J. M. De Villiers, P. N. Robinson, The Interval of Convergence and
Limiting Functions of a Hyperpower Sequence. The American Mathematical
Monthly, 93(1) (January 1986), 13-23. Perhaps the
nicest discussion on the iterated exponential, along with discussions
on the auxiliary equations and their use to establish that in (0, (1/e)^{e})
the infinite exponential is a two-cycle. Lambert's W function appears
in disguise in this article.
Roger Voles, An Exploration of Hyperpower Equations ^{n}x = ^{n}y, The Mathematical Gazette, 83(497) (July 1999), 210 - 215. A very nice article on some of the fundamental properties of
hyperexponentials with interesting identities on the derivative of the function ^{n}x
and an analysis on this function's behavior for x close to 1 and 0.
M. Ward, Note On The Iteration of Functions Of One Variable, Bulletin of the American Mathematical Society, 40 (1934), 688 - 690. Formulas for iterating a function E(x) continuously, such that E_{x+y}(x)=E_{x}(E_{x}(x)), based on Abel's functional equation.
M. Ward and F.B. Fuller, Continuous Iteration of Real Functions, Bulletin of the American Mathematical Society, 42 (1936), 383 - 386. Second paper on theoretical formulas for iterating a function E(x) continuously, again based on Abel's functional equation.
Gary Waters, problem 185, Mathematics and Computer Education, 18(1) (Winter 1984), 69-70. What is the largest a such that ^{n}a is a real number?
William V. Webb, Rooting Around For Roots, Mathematics and Computer Education, 24(3) (Fall 1990), 273. Solutions to the tetration equation z^{z}=1.
F. Woepcke, Note Sur L' Expression (((a^a)^a)^...)^a, Et Les Fonctions Inverses Correspondantes, Journal Fur Die Reine und Angewandte Mathematik [=Crelle's Journal], 42 (1851), 83 - 90. Various expressions and series expansions for the aforementioned expression (with lower/upper parenthesis orders) and its inverses.
Darren Wilson, Fractional Arithmetic, Journal of Undergraduate Mathematics, 12 (2)(Sep. 1980), 51-54. Must read for readers interested in extending exp^^n continuously.
E.M. Wright, A Class of Representing Functions, Journal of the London Mathematical Society, 29(1) (1954), 63-71. Characterizations and classifications of prime generating functions which generalize the Mills result [a^{3n}] is prime for all n in N.
E.M. Wright, On A Sequence Defined by A Non-Linear Recurrence Formula, Journal of the London Mathematical Society, 20(1) (1945), 68-73. Description of a function which grows slower than any repeated composition of log (or alternatively of a function which grows faster than any repeated composition of exp).
E.M. Wright, Iteration of The Exponential Function, Quarterly Journal of Mathematics (Oxford), 18 (1947), 228 - 235. Continuation of the previous reference, where the author exhibits analytic and continuous iteration of the function defined by the substitution (x|c*exp(x)). Must read for readers interested in extensions to tetration.
Shraga Yeshurun, Reverse Order Exponentiation, School Science and Mathematics, 89(2) (February 1989), 136 - 143. Detailed analysis of the orders of a^{b} and b^{a} with plots of the locus showing equality and inequalities.
Gene Zirkel, But Does it Converge?, Mathematics and Computer Education, 18(2) (Spring 1984), 153. Tetration problem for the specific value of x=sqrt(e*π).