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# Power Tower

The power tower of order is defined as

 (1)

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

 (2)

together with

 (3) (4)

Rucker (1995, p. 74) uses the notation

 (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 for , 2, ... for small .

 OEIS 1 A000027 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... 2 A000312 1, 4, 27, 256, 3125, 46656, ... 3 A002488 1, 16, 7625597484987, ... 4 1, 65536, ...

The following table gives for , 2, ... for small .

 OEIS 1 A000012 1, 1, 1, 1, 1, 1, ... 2 A014221 2, 4, 16, 65536, , ... 3 A014222 3, 27, 7625597484987, ... 4 4, 256, , ...

Consider and let be defined as

 (6)

(Galidakis 2004). Then for , is entire with series expansion:

 (7)

Similarly, for , is analytic for in the domain of the principal branch of , with series expansion:

 (8)

For , and ,

 (9)

For , and , and

 (10)
Min Max
 Min Max Re Im

The value of the infinite power tower , where is an abbreviation for , can be computed analytically by writing

 (11)

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

 (12)

Solving for gives

 (13)

where is the Lambert W-function (Corless et al. 1996). converges iff (; 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

 (14) (15)

(Vardi 1991).

The special value is given by

 (16) (17) (18)

(OEIS A077589 and A077590; Macintyre 1966).

The related function

 (19)

converges only for , that is, (OEIS A072364). The value it converges to is the inverse of which can be found by taking the logarithm of both sides of (19),

 (20)

rearranging to

 (21)

and then substituting to obtain

 (22)

Solving the resulting equation for then gives the partial solution

 (23)

which is valid for (i.e., ; OEIS A072364 and A073226). Taking then gives , where is the omega constant.

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

 (24)
Min Max

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

 (25)

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

 (26)

which has solution . At this point, the function takes on the value .

 (27)

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

 (28) (29) (30) (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."

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

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

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Power Tower

## Cite this as:

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