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
![a^^nk=a^^(n-1)[a^^n(k-1)]](/images/equations/PowerTower/NumberedEquation2.gif) |
(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)
|
 |
 |
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
![z^(z^(·^(·^·)))lnz=h(z)lnz=ln[h(z)].](/images/equations/PowerTower/NumberedEquation10.gif) |
(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)
|
 |
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.
Portions of this entry contributed by Ioannis Galidakis
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Function."
Adv. Comput. Math. 5, 329-359, 1996.
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."
J. reine angew. Math. 28, 49-52, 1844.
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Function to Infinite
Exponentials." Complex Variables Th. Appl. 49, 759-780, 2004.
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."
Int. J. Math. Educ. Sci. Technol. 20, 297-305, 1989.
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."
Proc. Amer. Math. Soc. 17, 67, 1966.
Mauerer, H. "Über die Funktion 
für
ganzzahliges Argument (Abundanzen)." Mitt. Math. Gesell. Hamburg 4,
33-50, 1901.
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Spindle."
Math. Mag. 69, 198-206, 1996.
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Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968.
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^
^
^
^...." Austral.
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Galidakis, Ioannis and Weisstein, Eric W. "Power Tower." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PowerTower.html
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2^2^2^2^2^2^2^2^2
