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Template:Tetration

(Redirected from Template:^^)
The {{tetration}} mathematical function template returns the tetration
 b ↑↑ h
, a power tower with nonnegative real as base
 b
, and with nonnegative integer in [0..32] as height
 h
, i.e.
${\displaystyle b{\uparrow \uparrow }h:=\underbrace {{b}^{{.\,}^{{.\,}^{{.\,}^{{b}^{{b}^{{b}^{b}}}}}}}} _{h},\quad b\in \mathbb {R} ^{+}\cup \{0\},\,h\in \mathbb {N} \cup \{0\},}$

otherwise returns an error message.

Beware: Calculated with the MediaWiki #expr parser function extension, which has a significand precision of 53 bits (52 explicitly stored) (approximately 16 decimal digits, 53 log10 2 ≈ 15.955).[1]

Usage

{{tetration|base|height}}

or

{{^^|base|height}}

Valid arguments

If neither the base nor the height is given, the empty product 1 is returned.

• First argument: base of power tower, as a nonnegative real number;
• Second argument: height of power tower, as a nonnegative integer in [0..32] (defaults to 1).

Examples

Examples with valid input

Returns
 b ↑↑ h
(or INF if overflow occurs, i.e. above ≈ 1.7976931348623157 x 10 308).[1]
Code Result Code Result
{{^^}} 1   {{^^|2}} 2
{{^^|0|0}} 1   {{^^|-1|0}} 1
{{^^|2|0}} 1   {{^^|2|1}} 2
{{^^|2|2}} 4   {{^^|2|3}} 16
{{^^|2|4}} 65536   {{^^|2|5}} INF
{{^^|3|3}} 7625597484987   {{^^|3|4}} INF

NOTE: {{^^|b|h}} for negative
 b
gives complex numbers (except for
 b =  −  1
) and we get NAN (Not A Number) since only real numbers are handled by the #expr: parser function.

{{#expr: (-1)^(-1) }} = -1
{{#expr: (-1.1)^(-1.1) }} = NAN
{{#expr: 1 / ( (-1.1)^(1.1) ) }} = NAN

Table of values of the tetration function (using {{^^|b|h}} template, except for manually entered last column entries)
${\displaystyle b\uparrow \uparrow 1}$ ${\displaystyle b\uparrow \uparrow 2}$ ${\displaystyle b\uparrow \uparrow 3}$ ${\displaystyle b\uparrow \uparrow 4}$ ${\displaystyle b\uparrow \uparrow 5}$ ${\displaystyle b\uparrow \uparrow 6}$ ${\displaystyle b\uparrow \uparrow 32}$ ${\displaystyle \lim _{h\to \infty }b\uparrow \uparrow h}$
-3 -0.037037037037037 NAN NAN NAN NAN NAN
-2.5 NAN NAN NAN NAN NAN NAN
-2 0.25 NAN NAN NAN NAN NAN
-1.5 NAN NAN NAN NAN NAN NAN
-1 -1 -1 -1 -1 -1 -1
-0.5 NAN NAN NAN NAN NAN NAN
0 1 0 1 0 1 1 undefined
0.1 0.79432823472428 0.16057272043212 0.69091922875998 0.20374209674284 0.62554405820608 0.4229202149354 0.39901297826025
0.2 0.7247796636777 0.31145890709838 0.60575856902197 0.37721845362499 0.5449235984186 0.46970963528893 0.46962192293561
0.3 0.69684530193595 0.43214988609837 0.59434417379411 0.48891128772632 0.55508396180265 0.52895696102551 0.52895644844295
0.4 0.69314484315515 0.5298707364733 0.6153797974711 0.56900457469329 0.59370446435584 0.58504317471451 0.58504317197440
0.5 0.70710678118655 0.61254732653607 0.65404086004207 0.63549784581337 0.64371864172287 0.64118574451391 0.64118574450499
0.6 0.73602192281783 0.68661684945978 0.70416577097103 0.69788152946118 0.700125434673 0.69953450919474 0.69953450919473
0.7 0.77905591267045 0.75739548555921 0.76326958144204 0.76167209672267 0.76210620802468 0.76201343081072 0.76201343081072
0.8 0.83651164207302 0.82972398762427 0.83098165747098 0.83074848273819 0.83079170890087 0.83078494866324 0.83078494866324
0.9 0.90953257608296 0.90861953913804 0.90870695076419 0.90869858185342 0.90869938310125 0.90869931309174 0.90869931309174
1 1 1 1 1 1 1 1
1.1 1.1105342410546 1.1116498000258 1.111768001503 1.1117805265341 1.1117818537373 1.1117820110418 1.11178201104184
1.2 1.244564747204 1.2547181707692 1.2570430412015 1.2575759820011 1.2576981823156 1.2577345413765 1.25773454137653
1.3 1.4064566732379 1.446293346286 1.4614888698918 1.4673271088311 1.469576402415 1.47098896009 1.47098896009015
1.4 1.6016928982022 1.7141634748254 1.780276017527 1.8203220817394 1.8450158019786 1.886662556747 1.88666330624633
1.5 1.8371173070874 2.1062033521489 2.3490053186119 2.5920257049075 2.8604414974606 INF ${\displaystyle +\infty }$
1.6 2.1212505710976 2.7101269178649 3.5743004801538 5.3652029167449 12.449305553378 INF ${\displaystyle +\infty }$
1.7 2.4646948994849 3.698164974975 7.1160358861268 43.639804309351 11395826597.324 INF ${\displaystyle +\infty }$

Table of values of the tetration function (using #expr: case by case, except for manually entered last column entries)
${\displaystyle b\uparrow \uparrow 1}$ ${\displaystyle b\uparrow \uparrow 2}$ ${\displaystyle b\uparrow \uparrow 3}$ ${\displaystyle b\uparrow \uparrow 4}$ ${\displaystyle b\uparrow \uparrow 5}$ ${\displaystyle b\uparrow \uparrow 6}$ ${\displaystyle b\uparrow \uparrow 32}$ ${\displaystyle \lim _{h\to \infty }b\uparrow \uparrow h}$
0 1 0 1 0 1   undefined
0.1 0.79432823472428 0.16057272043212 0.69091922875998 0.20374209674284 0.62554405820608   0.39901297826025
0.2 0.7247796636777 0.31145890709838 0.60575856902197 0.37721845362499 0.5449235984186   0.46962192293561
0.3 0.69684530193595 0.43214988609837 0.59434417379411 0.48891128772632 0.55508396180265   0.52895644844295
0.4 0.69314484315515 0.5298707364733 0.6153797974711 0.56900457469329 0.59370446435584   0.58504317197440
0.5 0.70710678118655 0.61254732653607 0.65404086004207 0.63549784581337 0.64371864172287   0.64118574450499
0.6 0.73602192281783 0.68661684945978 0.70416577097103 0.69788152946118 0.700125434673   0.69953450919473
0.7 0.77905591267045 0.75739548555921 0.76326958144204 0.76167209672267 0.76210620802468   0.76201343081072
0.8 0.83651164207302 0.82972398762427 0.83098165747098 0.83074848273819 0.83079170890087   0.83078494866324
0.9 0.90953257608296 0.90861953913804 0.90870695076419 0.90869858185342 0.90869938310125   0.90869931309174
1 1 1 1 1 1   1
1.1 1.1105342410546 1.1116498000258 1.111768001503 1.1117805265341 1.1117818537373   1.11178201104184
1.2 1.244564747204 1.2547181707692 1.2570430412015 1.2575759820011 1.2576981823156   1.25773454137653
1.3 1.4064566732379 1.446293346286 1.4614888698918 1.4673271088311 1.469576402415   1.47098896009015
1.4 1.6016928982022 1.7141634748254 1.780276017527 1.8203220817394 1.8450158019786   1.88666330624633
1.5 1.8371173070874 2.1062033521489 2.3490053186119 2.5920257049075 2.8604414974606   ${\displaystyle +\infty }$
1.6 2.1212505710976 2.7101269178649 3.5743004801538 5.3652029167449 12.449305553378   ${\displaystyle +\infty }$
1.7 2.4646948994849 3.698164974975 7.1160358861268 43.639804309351 11395826597.324   ${\displaystyle +\infty }$

Examples with invalid input

Returns an error message.

Code Result
{{^^|a|0}} Expression error: Unrecognized word "a".
{{^^|a|1}} Expression error: Unrecognized word "a".
{{^^|2i|3}} Expression error: Unrecognized word "i".
{{^^|2i+1|3}} Expression error: Unrecognized word "i".
{{^^|2|33}} Tetration error: Second argument (height) must be a nonnegative integer up to 32
{{^^|2|-1}} Tetration error: Second argument (height) must be a nonnegative integer up to 32
{{^^|2|0.5}} Tetration error: Second argument (height) must be a nonnegative integer

Test

The following few results for the tetration function (using {{^^|b|h}} template) fully agree (all 14 decimal digits) with Mathematica's Power @@ Table[b, {h}].

Table of values of the tetration function (using {{^^|b|h}} template) compared with Mathematica's Power @@ Table[b, {h}]
h {{^^|(1/e)^e|h}} Power @@ Table[(1/e)^e, {h}]
0 1 1
1 0.065988035845313 0.0659880358453125
2 0.83579318845359 0.835793188453588
3 0.10311386287497 0.103113862874971
4 0.75556267703185
5 0.12824249755517
6 0.70567587269525
7 0.14686728817137
8 0.6708386179955 0.670838617995497
9 0.16145506261086
10 0.644757912042
11 0.17331687438423
12 0.62430009159136 0.624300091591359
13 0.18322807430165
14 0.6077051016648
15 0.19168272746053 0.191682727460533
16 0.59389800988891 0.593898009888908
17 0.19901358938169
18 0.58218033596699
19 0.20545459299035 0.205454592990346
20 0.57207596716079
21 0.21117592903209
22 0.56324775217168
23 0.21630493523809
24 0.5554493934993 0.555449393499302
25 0.22093914155261
26 0.54849624036848
27 0.22515474374947 0.225154743749465
28 0.54224679029599
29 0.2290122927811
30 0.53659054896353
31 0.53659054896353
32 0.53659054896353 0.531439817592522