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)]](http://mathworld.wolfram.com/images/equations/PowerTower/NumberedEquation2.gif) |
(2)
|
together with
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
![a_(mn)=<span style=]() {1 if n=0; 1/(n!) if m=1; 1/nsum_(j=1)^(n)ja_(m,n-j)a_(m-1,j-1) otherwise " src="http://mathworld.wolfram.com/images/equations/PowerTower/NumberedEquation4.gif" style="height:114px; width:242px" /> |
(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)].](http://mathworld.wolfram.com/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 
(Vardi 1991).
The special value 
is given by
(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 
.
The indefinite integral
 |
(27)
|
cannot be expressed in terms of a finite number of elementary functions, but some interesting definite integrals of 
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 
is plotted above along the real line and in the complex plane, where it shows beautiful structure.
REFERENCES:
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as 
Tends to Infinity." Math. Mag. 69, 207-209, 1996.
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." J. reine angew. Math. 28, 49-52, 1844.
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." Proc. Amer. Math. Soc. 17, 67, 1966.
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Spindle." Math. Mag. 69, 198-206, 1996.
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^
^
^
^...." Austral. Math. Soc. Gaz. 22, 182-184, 1995.
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