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 .

	Sloane
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 .

	Sloane
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 .

The indefinite integral

(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.

REFERENCES:

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