Is the exact value of $^{\infty}i$ known?

Question

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?

Answer 1

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