Wikipedia's article on tetration has a table of successive tetrations of $i$ that seems to imply that $^{\infty}i$ converges, and my own experimentation seems to confirm it, but I'm suspicious of purely numerical results like that, and the article itself has no references for this portion of the text. Is the exact value known, or is the figure of approximately $0.4383 + 0.3606i$ quoted in that article the best that can be done?
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In particular, answer http://math.stackexchange.com/a/336719/442 – GEdgar Jun 08 '14 at 01:04
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The nine values given at your link do not convince me that there is a convergent sequence at hand, and the text of the article says the values are poorly understood because of "chaotic behavior". Perhaps supplying further details of your computation would serve to avoid having answers that are primarily opinion based. – hardmath Jun 08 '14 at 01:06
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You can express it in terms of the product log function or the W function but I do not believe it can be expressed in simpler terms.
$$^\infty i = \dfrac{\operatorname{W}(-\log(i))}{-\log(i)}$$
$$^\infty i = \dfrac{\operatorname{W}(-\dfrac{i\pi}{2})}{-\dfrac{i\pi}{2}}$$
$$^\infty i = \dfrac{2i\operatorname{W}(-\dfrac{i\pi}{2})}{{\pi}}$$
If you are looking for high precision here are quite a few digits:
0.43828293672703211162697516355126482426789735164639460360922124049579
1532222695687669172140538204075492890644949730472690155622229799337502
5387775370844821815525479373321921539182697140677674571053823184810431
8736151711542937512922834683049233094786721039036792854671915036644169
6820501953643922598255422410572811983883236582063517278832860195815821
9151757852903434752377692143667242793195747187086565651342378640597041
7031704917155871754239560487515280633220604694590842197513669600044237
7070278362945276669782676321876187877983416094346934324062726526111057
1910996909551063014323732411265280807518830658799194617928932234579175
6950458998761577661219337838077396509228047720993053001769398879127009
7311913355762542373341303549007838843875772889508936757546755444257410
2415495724306574265575879628845671143097585641264538990182672181646953
4235707398536098823919055907436046751729281595468439511503917043343406
8775347232309330126429298816061309035939131957693377286467883605775653
9543269740215931989514 +
0.3605924718713854859529405269060006538265770307860270047414512983804
6019521150773053292275414002568647305752347153230156355395645131352545
1234263983475453897610752205117036998817859580971727235510368739678103
6568663841983435705662647799770453833097652580992739352956370526318010
5315051408268510868922482075945182018173404095406652462981053982073252
3910676681626895560122807849790116505080069541500885972172329962269093
8349218406970951982501746810243726900438176347374219195323468384888206
4335479778259180719998992379659803021344715870308986727990327047853486
8150998816505411699680771508770897403344511273294484133290368712009732
9666482396630488861589975194441202008946901878558617505007480678060533
1114713764734213551742202000713023896428678830673939047111674355190209
0276110152621741098035587323806311144117903129785160906720449609288196
2275423849009782971403685939577380608290988507247630567665593437971777
9179146303786882801712133954612319878210988187576751492067175911679603
32758802375163700302099i
Brad
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The W function does give the exact value. Do you mean "cannot be simply expressed"? – Nathaniel Bubis Jun 08 '14 at 01:02
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