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.

  1. Problems, Notes, 7- 9, American Mathematical Monthly, 28(3) (March 1921), 140 - 143. A discussion on the equation xy = yx along with some short notes on its parametrization.
  2. 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 mn = nm.
  3. Jeffrey M. Alden , A Closer Look at xxx... . Journal of Undergraduate Mathematics, 11(2) (Sep. 1979), 63-66. Parametrization of the locus of the period 2 iterated exponential.
  4. 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.
  5. Arnold O. Allen, eπ or πe?. Journal of Recreational Mathematics, 2(4), (October 1969), 255 - 256. Elementary analysis of when the different exponentials are the same.
  6. 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.
  7. Joel Anderson, Iterated Exponentials, The Mathematical Association of America, 111 Monthly (October 2004), 668 - 679. A short presentation on the sequence a, aa, aaa.
  8. 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).
  9. Fashid Arjomandi, Problem 27.1, Mathematical Spectrum, 27(3) (1994/1995), 68. Number theoretic tetration problem about divisibility.
  10. R.C. Archibald, R. Clare, Problem 884, Mathematics of Computation, 5(35), (July 1951), 140. Solutions to the first and second auxiliary tetration equations.
  11. R.C. Archibald, R. Clare, Problem 136, Mathematics of Computation, 6(39) (July 1952), 204. Solution of y=ey.
  12. J. Marshall Ash, The Limit of x^x^...^x as x Tends to Zero, Mathematics Magazine 69 (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.
  13. H.D' Assumpcao, G. Grossley, R.J. Armstrong, Note 2891. Iterated Exponentials, The Mathematical Gazette 44 #348(May 1960), 126-127.
  14. F. Azarpanah, Proof Without Words: Convergence of a Hyperpower Sequence, Mathematics Magazine 77 (2004), 393.
  15. 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,
  16. 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) = cz, along with some very nice diagrams that show the domains of convergence on the complex plane similar to the fractal on article 1.
  17. 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.
  18. 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,...}.
  19. 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 x1/x decompositions of the iterate bases, similar to the author's analysis on article 1.
  20. Morris H. Barshinger , A Look at xxx... . Journal of Undergraduate Mathematics, 5(1) (March 1972), 37-41.
  21. 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.
  22. 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.
  23. W. W. Beman, Problem 389. American Mathematical Monthly, 21(1) (January 1914), 23. A series expansion of eex.
  24. 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).
  25. Bob Bertuello, An Infinite Exponential. Mathematical Spectrum, 27(1) (1994/95), 22. One more reader discovers the fascinating convergence of the iterated exponential.
  26. 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.
  27. 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.
  28. 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=xx.
  29. 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, expn(x).
  30. 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.
  31. N.D. Hayes, The Roots of The Equation x=(c*exp)nx And The Cycles of The Substitution (x|cex). Quarterly Journal of Mathematics (Oxford), 3(2) (1952), 81 - 90. Very nice exposition for the roots of the equation x=(c*exp)nx, 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.
  32. 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.
  33. 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.
  34. Florian Cajori, History of the Exponential and Logarithmic Concepts (part 2). American Mathematical Monthly, 20(2), (February 1913), 35 - 47. Part 2 of above.
  35. Errata in the February Issue. American Mathematical Monthly, 20(3) (March 1913), 104. Errata of the above.
  36. Florian Cajori, History of the Exponential and Logarithmic Concepts (part 3). American Mathematical Monthly, 20(3) (March 1913), 75 - 84. Part 3 of above.
  37. Florian Cajori, History of the Exponential and Logarithmic Concepts (part 4). American Mathematical Monthly, 20(4) (April 1913), 107 - 117. Part 4 of above.
  38. Florian Cajori, History of the Exponential and Logarithmic Concepts (part 5). American Mathematical Monthly, 20(5) (May 1913), 148 - 151. Part 5 of above.
  39. Florian Cajori, History of the Exponential and Logarithmic Concepts (part 6). American Mathematical Monthly, 20(6) (June 1913), 173 - 182. Part 6 of above.
  40. Florian Cajori, History of the Exponential and Logarithmic Concepts (part 7). American Mathematical Monthly, 20(7) (September 1913), 205 - 210. Part 7 of above.
  41. S. Camosso, On a kind of Gaussian integral with the square on the left. ARCHIMEDE MATHEMATICAL JOURNAL, 8(1) (Spring 2021), 11-18. New identities and inequalities for tetrational integrals.
  42. 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.
  43. R. D. Carmichael, Problem 275. American Mathematical Monthly, 14(2) (February 1907), 27. Solution to equations xy-yx=0 and y-x=a*(a+1)1/a.
  44. Arthur Latham Baker, Functional Exponents. School Science and Mathematics, 8(3) (March 1908), 225 - 227. Elementary discussion on what fa(x) means for integer and fractional a.
  45. E. Barbette, Des Progressions Logarithmiques. Mathesis Requeil Mathematique, 8(2) (1898), 135 - 137. Sequences whose exponents reduce to geometric progressions.
  46. 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.
  47. 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.
  48. 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.
  49. Michael Brozinsky, Problem 4394. School Science and Mathematics, 93(6) (October 1993), 340. A finite tetration problem and a related inequality.
  50. R. A. Bruce (student), General repeated exponentiation. The American Mathematical Monthly, 71(8) (October 1964), 759 - 761. Bounds of convergence for iterated exponential.
  51. 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.
  52. E. P. Bugdanoff, Problem 3515 [1931,462]. American Mathematical Monthly, 39(9) (November 1932), 552 - 555. Elementary analysis of the equation 2x=4*x.
  53. 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 xy = yx.
  54. F. A. Butter, A note on the equation xy=yx. American Mathematical Monthly, 46(6), (June/July 1939), 316 - 317. Short note on the real solutions of the aforementioned equation.
  55. W. D. Cairns, Problem 422. The American Mathematical Monthly, 22(4) (April 1915), 133. The solution to the equation xx*sqrt(x)=(x*sqrt(x))x.
  56. 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 xy - yx.
  57. 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.
  58. 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 an+1=aan.
  59. 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].
  60. 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.
  61. 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.
  62. Charles C. Cross, Problem 93. The American Mathematical Monthly, 6(8/9) (August/September 1899), 199-201. System of two equations of the form xy+/-yx=c.
  63. H. Martyn Cundy, x^y=y^x: An investigation. The Mathematical Gazette, 71(456) (June 1987), 131-135.
  64. 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.
  65. Ed. Delville, Nombres Egaux a Leurs Logarithmes. Mathesis, 4(2) (1912), 264. Discussion on when a number is equal to its logarithm.
  66. 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.
  67. 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.
  68. 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)=expn(fn(z)), with fn being the iterates of exp.
  69. 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.
  70. 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 xy=yx.
  71. Carlton Lee Evans, Interesting Logs, Reader's Reflection Column, Mathematics Teacher, 72(7) (October 1979), 489. Elementary analysis of the function x1/x.
  72. C. J. Everett, Jr., An Exponential Diophantine Equation, American Mathematical Monthly, 43(4), (April 1936), 229 - 230. Rational solutions of the equation xy=yx.
  73. 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.
  74. 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.
  75. 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.
  76. Philip Franklin, Discussions, Relating to the Real Locus Defined by the Equation xy = yx. The American Mathematical Society, 24(3) (March 1917), 137. Discussion similar to the above, but not as elaborate.
  77. 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.
  78. Daniel Geisler, Tetration.org Page. Lots of work on the dynamics of the map z|->cz, 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|->cz 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.
  79. 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.
  80. 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)=cz.
  81. 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.
  82. 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.
  83. 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 xx=c.
  84. F. Gerrish, a^b=b^a: the positive integer solution. The Mathematical Gazette, 76(#477) (Nov. 1992), 403.
  85. 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.
  86. Richard A. Gibbs, Problem 4334. School Science and Mathematics, 92(5) (May/June 1992), 292-293. Intersection of the graphs of ax and loga(x).
  87. Jekuthiel Ginsburg, Iterated Exponentials. Scripta Mathematica II, (1945), 340 - 353. Relationships on the coefficients of the series of the iterated exponential function expn(x).
  88. 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.
  89. Peter Goodstein, Note 2803 Limits of iterated logarithmic functions. The Mathematical Gazette, 42(#342) (Dec. 1958), 295-296.
  90. R.L. Goodstein, Note 2804. The system of equations bx=x1, bx1=x2,..., bxn=x. The Mathematical Gazette, 42(#342) (Dec. 1958), 296-299.
  91. R.L. Goodstein, The Equation ab=ba. The Mathematical Gazette, 28(279) (May 1944), 76. Quick and elegant analysis of both integral and non-integral solutions of the infamous equation.
  92. 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.
  93. 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.
  94. 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 loga(b)< b for all positive b?
  95. 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.
  96. 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!).
  97. 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.
  98. David Hammer, Problem 171, Parenthetical Roots, Mathematics and Computer Education, 17(1) (Winter 1983), 73. Finite tetration problem for sqrt(2).
  99. Alvin Hausner, Problem E 1474 [1961,573], The Equation mnm = nmn. The American Mathematical Monthly, 69(2) (February 1962), 169. Shows that the aforementioned equation has no solution in integers m <> n.
  100. Alvin Hausner, Algebraic Number Fields and the Diophantine Equation mn = nm. 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.
  101. 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.
  102. 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 xy = yz, with x and y transfinite ordinals.
  103. J. Higgins, On Note 3232, The Mathematical Gazette 55#391 (Feb. 1971), 63-64.
  104. Solomon Hurwitz, On the Rational Solutions of mn= nm, 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 mn = nm.
  105. 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.
  106. Nishizawa, Kiyoko, Convergence of a^a^a^... and bifurcation of dynamical systems. Sci. Rep. Res. Inst. Eng., 11(1988), 54-60.
  107. Y. S. Kupitz, H Martini, On The Equation xy = yx, Elemente der Mathematik, 55 (2000) 95 -101. A quite thorough investigation of the above equation with a parametrization of its non-trivial solutions.
  108. Honsta. W. Labbers, Jr., Problem 420, Nieuw Archief voor WisKunde, 3(24) (1976), 207-210. A number-theoretical problem related to the Ackermann function.
  109. 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.
  110. L. J. Lander, Problem E 1124 [1954,423], The American Mathematical Monthly, 62(2) (February 1955), 124 - 125. Solutions of the equation mn=nm, for integer m and n.
  111. 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->ax.
  112. J. Lense, 3053. The American Mathematical Monthly, 31(10) (December 1924), 500 - 501. A short problem on the real infinite exponential iteration.
  113. H.W. Lenstra, Jr., 3053. Problem 566, Nieuw Archief voor WisKunde, 3(28) (1980), 300-302. Short problem on the growth of the function 22n.
  114. J.E. Littlewood (ref), Note 2285, A Note on Multiple Exponentials, The Mathematical Gazette 36#316 (May 1952), 127 - 128.
  115. J.E. Littlewood (ref), Note 2517, On Note 2285, The Mathematical Gazette 39#328 (May 1955), 141 - 142.
  116. 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^....
  117. 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 ax = x, which bares interesting connections with the function f(x) = x1/x as it was investigated in article 1.
  118. J. Van De Lune, Problem 899, Nieuw Archief voor Wiskunde, 4(13) (1995), 251 - 252. Problem related to iterated exponentiation sequence.
  119. J. Macdonnell, Some Critical Points on the Hyperpower Function nx = 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).
  120. 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^....).
  121. Warren B. Manhard 2d, Is Exponentiation Commutative? Mathematics Teacher, 74(1) (January 1981), 56 - 60. Exploration of the equation ab=ba, with graphical plots of the solution locus for some reals.
  122. 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)=bx and g(x)=logb(x).
  123. Patrick J. McCarthy, Decompositions of Functions, Functional nth roots of unity (in functional equations). Mathematical Gazette, 64(428) (June 1980), 107-115.
  124. J.S. Griffith, Decompositions of Functions, Some approaches to the general problem (in functional equations). Mathematical Gazette, 64(428) (June 1980), 116-120.
  125. 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) = x1/x  and the associated functions for some cases on the iterated exponentiation.
  126. E. J. Moulton, The Real Function Defined by xy = yx. 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.
  127. Joseph Morton, Graphical Study of xx=y. The Mathematical Student Journal, 9(2) (January 1962), 5. Analysis of the equation xx=y using complex numbers.
  128. 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.
  129. 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)) ~ ex (in terms of order of magnitude)?.
  130. Sam Newman (proposer), Problems and Solutions , Mathematics Magazine, 40(1967), 283-284. Exponential Derivative.
  131. Ivan Niven, Which is Larger, eπ or πe?, 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.
  132. 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.
  133. Jean J, Petersen and Peter Hilton, exp(π) > πe? Readers Reflections Column, Mathematics Teacher, 74(7) (October 1981), 501 - 502. A discussion of the "Is Exponentiation Commutative" article on the same magazine.
  134. 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)=ef(x).
  135. Louis O'Shaughnessy, Problem 433. American Mathematical Monthly, 26(1) (January 1919), 37-39. Fractional derivative problem.
  136. Stanley Rabinowitz, Problem 191, Exponential Roots, Mathematics and Computer Education, 18(2) (Spring 1984), 150-151. Solutions of the equation xx=c for various c.
  137. W. R. Ransom, Problem E 3 [1932,489]. American Mathematical Monthly, 40(2) (February 1933), 113. The equation y=xx again.
  138. Dave L. Renfro, Exponential and Logarithmic Commutativity, The Mathematics Teacher, 91 (1998), 275, 362. A nice discussion on the analytic solvability of equations, exp(10x) = 10exp(x), ln(log(x)) = log(ln(x)) and extending the ideas to the more general classes of equations, ab^x = ba^x and loga(logb(x)) = logb(loga(x)).
  139. 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.
  140. R. Robinson Rowe, The Mutuabola. Journal of Recreational Mathematics, 3(3) (July 1970), 176 - 178. Analysis of the solutions of the equation xy=yx.
  141. 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.
  142. Daihachiro Sato, Shorter Notes, Algebraic Solution of xy = yx (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.
  143. J. Scheffer, Problem 307. The American Mathematical Monthly, 16(2) (February 1909), 32 Find x and y if yx=2 and xy=3.
  144. 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 xy=yx.
  145. 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.
  146. 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.
  147. Thomas W. Shilgalis, Graphical Solution of The Equation ab = ba. The Mathematics Teacher, 66(3) (Mar. 1973), 235. Short paper which arose from the question on whether eπ <?> πe.
  148. Steven L. Siegel, Exponential Equations. Reader Reflections Columnm, Mathematics Teacher, 100(4) (November 2006), 238. Solutions of the equation xa=ax.
  149. Wactaw Sierpinski, (pp. 106-109 in) Elementary Theory of Numbers. Monografie Matematyczne, 42 (1964), 106-109.
  150. David J. Silverman, What's the Limit? Journal of Recreational Mathematics, 4(2) (April 1971), 144. Is the limit of xx... equal to 2 or 4?
  151. Man-Keung Siu, An Interesting Exponential Equation Mathematical Gazette, 60(413) (Oct. 1976), 213-215. The equation: x1^x2=x2^x3=...=xn^x1.
  152. H. L. Slobin, The solutions of xy = yx, 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.
  153. G. P. Speck, ab Versus ba. School Science and Mathematics, 65(6) (June 1965), 489 - 490. The two different powers compared.
  154. V. M. Spunar, Problem 430. American Mathematical Monthly, 26(9) (November 1919), 415. Solution of the equations xy+yx=x*y and xy+yx=x+y.
  155. 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.
  156. L. John Stroud, On x=ax. The Mathematical Gazette, 53(385) (Oct. 1969), 289-293.
  157. Karl R. Stromberg. (pp. 185-186 of) Introduction to Classical Real Analysis, Wadswath 1981, 185-186.
  158. 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.
  159. 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.
  160. 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).
  161. 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=x1/x, and subsequent analysis.
  162. 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.
  163. H. S. Uhler, On the Numerical Value of ii. American Mathematical Monthly, 28(3), (March 1921), 114 - 121. Numerical estimates for various fundamental exponential constants, followed by historical notes on the relation ii=exp(-π/2) by R. C. Archibald.
  164. John T. Varner III, Comparing ab and ba Using Elementary Calculus. Two-Year College Mathematics Journal (now College Mathematics Journal), 7(4) (December 1976), 46. Short paper on determination of ab <?> ba, with a corollary for e and π.
  165. Glenn T. Vickers, Experiments With Infinite Exponents. Mathematical Spectrum, 27(2) (1994/95), 34. Discussion on "An Infinite Exponential" on the same magazine.
  166. Glenn T. Vickers, More About An Infinite Exponential. Mathematical Spectrum, 27(3) (1994/95), 54 - 56. Elementary analysis of some iterated exponential sequences.
  167. 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.
  168. Roger Voles, An Exploration of Hyperpower Equations nx = ny, 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 nx and an analysis on this function's behavior for x close to 1 and 0.
  169. 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 Ex+y(x)=Ex(Ex(x)), based on Abel's functional equation.
  170. 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.
  171. Gary Waters, problem 185, Mathematics and Computer Education, 18(1) (Winter 1984), 69-70. What is the largest a such that na is a real number?
  172. William V. Webb, Rooting Around For Roots, Mathematics and Computer Education, 24(3) (Fall 1990), 273. Solutions to the tetration equation zz=1.
  173. R.O. Webber, J. Roumeliotis, iii..., Aust. Math. Soc. Gazette, 22(4) (Oct. 1995), 1-1.
  174. 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.
  175. Darren Wilson, Fractional Arithmetic, Journal of Undergraduate Mathematics, 12 (2)(Sep. 1980), 51-54. Must read for readers interested in extending exp^^n continuously.
  176. 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 [a3n] is prime for all n in N.
  177. 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).
  178. 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.
  179. Shraga Yeshurun, Reverse Order Exponentiation, School Science and Mathematics, 89(2) (February 1989), 136 - 143. Detailed analysis of the orders of ab and ba with plots of the locus showing equality and inequalities.
  180. 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).