Power Tower

The power tower of order k is defined as


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


together with


Rucker (1995, p. 74) uses the notation


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]


  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.

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.

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

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


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:


For m in N, and x in R,


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

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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


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


Solving for h(z) gives


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)


(Vardi 1991).

The special value h(i) is given by

 approx 0.438283+0.3605924i

(OEIS A077589 and A077590; Macintyre 1966).


The related function


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),


rearranging to


and then substituting to obtain


Solving the resulting equation for x then gives the partial solution


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

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


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


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

The indefinite integral


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


(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."


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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Referenced on Wolfram|Alpha

Power Tower

Cite this as:

Galidakis, Ioannis and Weisstein, Eric W. "Power Tower." From MathWorld--A Wolfram Web Resource.

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