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Power Tower

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The power tower of order k is defined as

a^^k=a^(a^(·^(·^(·^a))))_()_(k),
(1)

where ^ is Knuth up-arrow notation (Knuth 1976), which in turn is defined by

a^^nk=a^^(n-1)[a^^n(k-1)]
(2)

together with

a^k=a^k
(3)
a^^n1=a.
(4)

Rucker (1995, p. 74) uses the notation

^ka=a^(a^(·^(·^(·^a))))_()_(k),
(5)

and refers to this operation as "tetration."

A power tower can be implemented in the Wolfram Language as 

  PowerTower[a_, k_Integer] := Nest[Power[a, #]&, 1, k]

or 

  PowerTower[a_, k_Integer] := Power @@ Table[a, {k}]

The following table gives values of a^(a^(·^(·^(·^a))))_()_(n) for a=1, 2, ... for small n.

nOEISa^(a^(·^(·^(·^a))))_()_(n)
1A0000271, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
2A0003121, 4, 27, 256, 3125, 46656, ...
3A0024881, 16, 7625597484987, ...
41, 65536, ...

The following table gives a^(a^(·^(·^(·^a))))_()_(n) for n=1, 2, ... for small a.

aOEISa^(a^(·^(·^(·^a))))_()_(n)
1A0000121, 1, 1, 1, 1, 1, ...
2A0142212, 4, 16, 65536, 2.00×10^(19728), ...
3A0142223, 27, 7625597484987, ...
44, 256, 1.34×10^(154), ...

Consider z^(z^(·^(·^(·^z))))_()_(m) and let a_(mn) be defined as

a_(mn)={1 if n=0; 1/(n!) if m=1; 1/nsum_(j=1)^(n)ja_(m,n-j)a_(m-1,j-1) otherwise
(6)

(Galidakis 2004). Then for m in N, (e^z)^((e^z)^(·^(·^(·^((e^z))))))_()_(m) is entire with series expansion:

(e^z)^((e^z)^(·^(·^(·^((e^z))))))_()_(m)=sum_(n=0)^m((n+1)^n)/((n+1)!)z^n+sum_(n=m+1)^inftya_(mn)z^n.
(7)

Similarly, for m in N, z^(z^(·^(·^(·^z))))_()_(m) is analytic for z in the domain of the principal branch of lnz, with series expansion:

z^(z^(·^(·^(·^z))))_()_(m)=sum_(n=0)^m((n+1)^n)/((n+1)!)ln^nz+sum_(n=m+1)^inftya_(mn)ln^nz.
(8)

For m in N, and x in R,

int(e^x)^((e^x)^(·^(·^(·^((e^x))))))_()_(m)dx=sum_(n=0)^m((n+1)^(n-2))/(n!)x^(n+1) +sum_(n=m+1)^infty(a_(mn))/(n+1)x^(n+1).
(9)

For m in N, and x>0, and b(n+1,x)=Gamma(n+1,-ln(x))

intx^(x^(·^(·^(·^x))))_()_(m)dx=sum_(n=0)^m((-1)^n(n+1)^(n-1))/(n!)b(n+1,x) +sum_(n=m+1)^infty(-1)^na_(mn)b(n+1,x).
(10)
PowerTowerH
PowerTowerHReIm
PowerTowerHContours

The value of the infinite power tower h(z)=z^^infty=z^(z^(·^(·^·))), where z^(z^z) is an abbreviation for z^((z^z)), can be computed analytically by writing

z^(z^(·^(·^·)))=h(z)
(11)

taking the logarithm of both sides and plugging back in to obtain

z^(z^(·^(·^·)))lnz=h(z)lnz=ln[h(z)].
(12)

Solving for h(z) gives

h(z)=-(W(-lnz))/(lnz),
(13)

where W(z) is the Lambert W-function (Corless et al. 1996). h(z) converges iff e^(-e)<=x<=e^(1/e) (0.0659<=x<=1.4446; OEIS A073230 and A073229), as shown by Euler (1783) and Eisenstein (1844) (Le Lionnais 1983; Wells 1986, p. 35).

Knoebel (1981) gave the following series for h(z)

h(z)=sum_(n=0)^(infty)((n+1)^nln^nz)/((n+1)!)
(14)
=1+lnz+(3^2(lnz)^2)/(3!)+(4^3(lnz)^3)/(4!)+...
(15)

(Vardi 1991).

The special value h(i) is given by

i^(i^(·^(·^·)))=-(W(-lni))/(lni)
(16)
=(2i)/piW(-1/2pii)
(17)
approx 0.438283+0.3605924i
(18)

(OEIS A077589 and A077590; Macintyre 1966).

PowerTowerG

The related function

g(x)=x^((1/x)^((1/x)^...))
(19)

converges only for x>=e^(-1/e), that is, x>=0.692 (OEIS A072364). The value it converges to is the inverse of x^x which can be found by taking the logarithm of both sides of (19),

lng=(1/x)^((1/x)^((1/x)^...))lnx,
(20)

rearranging to

x^((1/x)^((1/x)^...))lng=lnx,
(21)

and then substituting to obtain

glng=lnx.
(22)

Solving the resulting equation for x then gives the partial solution

g(x)=(lnx)/(W(lnx)),
(23)

which is valid for e^(-1/e)<=x<=e^e (i.e., 0.692<x<15.154; OEIS A072364 and A073226). Taking x=e then gives 1/W(1), where W(1) is the omega constant.

A continued fraction due to Khovanskii (1963) for the single iteration of g(x) is given by

x^(1/x)=1+(2(x-1))/(x^2+1-((x^2-1)(x-1)^2)/(3x(x+1)-((4x^2-1)(x-1)^2)/(5x(x+1)-((9x^2-1)(x-1)^2)/(7x(x+1)-...)))).
(24)
XToTheX
XToTheXReIm
XToTheXContours

The function z^z is plotted above along the real line and in the complex plane. It has series expansion

x^x=1+xlnx+1/2x^2(lnx)^2+1/6x^3(lnx)^3+...
(25)

(Trott 2004, p. 59). It has a minimum where

d/(dx)x^x=x^x(1+lnx)=0,
(26)

which has solution x=1/e. At this point, the function takes on the value e^(-1/e).

The indefinite integral

intx^xdx
(27)

cannot be expressed in terms of a finite number of elementary functions, but some interesting definite integrals of x^x are

int_0^1x^xdx=sum_(n=1)^(infty)((-1)^(n+1))/(n^n)
(28)
=0.7834305107...
(29)
int_0^1x^(-x)dx=sum_(n=1)^(infty)1/(n^n)
(30)
=1.2912859971...
(31)

(OEIS A083648 and A073009; Spiegel 1968; Abramowitz and Stegun 1972; Havil 2003, pp. 44-45; Borwein et al. 2004, p. 5). Borwein et al. (2004, pp. 5 and 44) call these two integrals "a sophomore's dream."

PowerTower3
PowerTower3ReIm
PowerTower3Contours

The function z^(z^z) is plotted above along the real line and in the complex plane, where it shows beautiful structure.

See also

Ackermann Function, Exponential Factorial, Exponential Function, Fermat Number, Joyce Sequence, Knuth Up-Arrow Notation, Lambert W-Function, Mills' Constant, MRB Constant, Nested Radical, Omega Constant, Power, Sophomore's Dream, Steiner's Problem

Portions of this entry contributed by Ioannis Galidakis

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Ash, J. M. "The Limit of x^(x^(·^(·^(·^x)))) as x Tends to Infinity." Math. Mag. 69, 207-209, 1996.Baker, I. N. and Rippon, P. J. "Convergence of Infinite Exponentials." Ann. Acad. Sci. Fennicæ Ser. A. I. Math. 8, 179-186, 1983.Baker, I. N. and Rippon, P. J. "Iteration of Exponential Functions." Ann. Acad. Sci. Fennicæ Ser. A. I. Math. 9, 49-77, 1984.Baker, I. N. and Rippon, P. J. "A Note on Complex Iteration." Amer. Math. Monthly 92, 501-504, 1985.Barrow, D. F. "Infinite Exponentials." Amer. Math. Monthly 43, 150-160, 1936.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 61-62, 2004.Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; and Knuth, D. E. "On the Lambert W Function." Adv. Comput. Math. 5, 329-359, 1996.Creutz, M. and Sternheimer, R. M. "On the Convergence of Iterated Exponentiation, Part I." Fib. Quart. 18, 341-347, 1980.Creutz, M. and Sternheimer, R. M. "On the Convergence of Iterated Exponentiation, Part II." Fib. Quart. 19, 326-335, 1981.de Villiers, J. M. and Robinson, P. N. "The Interval of Convergence and Limiting Functions of a Hyperpower Sequence." Amer. Math. Monthly 93, 13-23, 1986.Eisenstein, G. "Entwicklung von alpha^(alpha^(alpha^...))." J. reine angew. Math. 28, 49-52, 1844.Elstrodt, J. "Iterierte Potenzen." Math. Semesterber. 41, 167-178, 1994.Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29-51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. pp. 350-369.Finch, S. R. "Iterated Exponential Constants." §6.11 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 448-452, 2003.Galidakis, I. N. "On An Application of Lambert's W Function to Infinite Exponentials." Complex Variables Th. Appl. 49, 759-780, 2004.Ginsburg, J. "Iterated Exponentials." Scripta Math. 11, 340-353, 1945.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Khovanskii, A. N. The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory. Groningen, Netherlands: P. Noordhoff, 1963.Knoebel, R. A. "Exponentials Reiterated." Amer. Math. Monthly 88, 235-252, 1981.Knuth, D. E. "Mathematics and Computer Science: Coping with Finiteness. Advances in our Ability to Compute are Bringing us Substantially Closer to Ultimate Limitations." Science 194, 1235-1242, 1976.Länger, H. "An Elementary Proof of the Convergence of Iterated Exponentials." Elem. Math. 51, 75-77, 1996.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, pp. 22 and 39, 1983.Macdonnell, J. "Some Critical Points on the Hyperpower Function ^nx=x^(x^(·^(·^(·^x))))." Int. J. Math. Educ. Sci. Technol. 20, 297-305, 1989.Macintyre, A. J. "Convergence of i^(i^(·^(·^·)))." Proc. Amer. Math. Soc. 17, 67, 1966.Mauerer, H. "Über die Funktion x^(x^...) für ganzzahliges Argument (Abundanzen)." Mitt. Math. Gesell. Hamburg 4, 33-50, 1901.Meyerson, M. D. "The x^x Spindle." Math. Mag. 69, 198-206, 1996.Rippon, P. J. "Infinite Exponentials." Math. Gaz. 67, 189-196, 1983.Rucker, R. Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1995.Sloane, N. J. A. Sequences A072364, A073009, A073226, A073229, A073230, A077589, A077590, and A083648 in "The On-Line Encyclopedia of Integer Sequences."Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. https://www.mathematicaguidebooks.org/.Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 11-12 and 226-229, 1991.Weber, R. O. and Roumeliotis, J. "i^i^i^i^...." Austral. Math. Soc. Gaz. 22, 182-184, 1995.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 35, 1986.

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Power Tower

Cite this as:

Galidakis, Ioannis and Weisstein, Eric W. "Power Tower." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PowerTower.html

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