The power tower of positive integer order 
consists of 
copies of a base 
, with exponentiation associated
from the right. It is denoted by
 |
(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.svg) |
(2)
|
for integers 
and 
, together with
 |  |  |
(3)
|
 |  |  |
(4)
|
Rucker (1995, p. 74) uses the notation
 |
(5)
|
and refers to this operation as "tetration."
For nonnegative integer 
, with the height-zero tower defined to be 1, a power tower
can be implemented in the Wolfram Language
as
PowerTower[a_, k_Integer] /; k >= 0 := Nest[Power[a, #]&, 1, k]
or
PowerTower[a_, k_Integer] /; k >= 0 := 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 
, where 
is the incomplete
gamma function,
 |
(10)
|
When the defining sequence converges, the infinite power tower 
,
where 
is an abbreviation for 
,
satisfies the fixed point equation obtained by writing
 |
(11)
|
For positive real number 
, taking the logarithm of both
sides gives
![z^(z^(·^(·^·)))lnz=h(z)lnz=ln[h(z)].](/images/equations/PowerTower/NumberedEquation10.svg) |
(12)
|
Solving for 
gives
 |
(13)
|
where 
is the principal branch of the Lambert
W-function (Corless et al. 1996), and the removable value at 
is 
. For positive real number 
, the defining sequence
for 
converges iff 
(
; OEIS A073230
and A073229), as shown by Euler (1783) and
Eisenstein (1844) (Le Lionnais 1983; Wells 1986, p. 35).
For 
, the odd-height and even-height
towers converge to distinct limits 
and 
,
respectively, with 
.
They satisfy 
and 
, so both satisfy 
(Knoebel 1981, pp. 241-242). At 
, the two limits coincide
at 
.
Knoebel (1981) gave the following series for 
, with the principal branch
of 
and 
 |  |  |
(14)
|
 |  |  |
(15)
|
(Vardi 1991).
The special value 
is given by
 |  |  |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
(OEIS A077589 and A077590; Macintyre 1966).
The related function
 |
(19)
|
converges for positive real number 
iff 
. Its limit is the inverse
function of 
restricted to 
,
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 
gives
 |
(23)
|
which is valid for 
(approximately 
;
OEIS A072364 and A073226),
with the removable value 
.
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). For 
, 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.
as 
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^
^
^
^...." Austral. Math. Soc. Gaz. 22, 182-184,
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