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 Complex Tetration Balarka Sen Junior Fellow Posts: 25 Threads: 7 Joined: Feb 2013 02/17/2013, 01:27 PM Hi, I was reading the article and found that the author describes a method to extend the tetration to reals. It uses the recursive relation and the differentiability requirement but a piecewise argument, hence not following the continuity requirement(?). It was a nice method, I am thinking about using it in practice. But I wonder if there are any way to extend the a^^x to complex values of a and x. Do we already have such analytic continuation? I am mainly interested in the recursion and differentiability requirement. Balarka . bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 02/18/2013, 09:33 PM (02/17/2013, 01:27 PM)Balarka Sen Wrote: I was reading the article Which article are you referring to? tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 02/18/2013, 11:20 PM (02/18/2013, 09:33 PM)bo198214 Wrote: (02/17/2013, 01:27 PM)Balarka Sen Wrote: I was reading the article Which article are you referring to? Yes I was wondering the same. As far as I know , we have not yet have " the " article. Btw I do not know what to think of those dropped conditions :/ mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 02/18/2013, 11:40 PM (This post was last modified: 02/19/2013, 06:03 AM by mike3.) (02/17/2013, 01:27 PM)Balarka Sen Wrote: Hi, I was reading the article and found that the author describes a method to extend the tetration to reals. It uses the recursive relation and the differentiability requirement but a piecewise argument, hence not following the continuity requirement(?). It was a nice method, I am thinking about using it in practice. But I wonder if there are any way to extend the a^^x to complex values of a and x. Do we already have such analytic continuation? I am mainly interested in the recursion and differentiability requirement. Balarka . What were you reading? I can think of the Wikipedia article, Hooshmand's paper, and Andrew Robbins' paper. These all mention a piecewise method, and the Wiki article mentions a "differentiability requirement". As for the question of how to extend to complex values, there are a few that seem to give the same result, namely Kneser's method and the Cauchy integral. There is also the "continuum sum", the "matrix method" and the above mentioned Robbins' method (which you may be referring to). All five of these appear to yield the same result, but it is not proven, as far as I can tell. In theory, these should be able to define $^z a$ for any complex a and z. But the currently-available implementations only work for more limited ranges of bases a. (Ed: The Cauchy integral may not work for bases on the real interval $1 < a < e^{1/e}$, at least if we want it to be analytically compatible with the rest of the bases. Tetration, it seems, as given by these methods has branch points at $e^{1/e}$, $1$, and $0$ (and possibly more) in the base. A cut is taken on $(-oo, e^{1/e}]$ to yield a principal branch, and we can define in there (at the cost of discontinuity) via the "continuity from above" convention. Tetration is not real-valued on this interval for non-integer "heights". So far, I'm not sure any luck has been had at solving the Cauchy integral equation on a complex base of any sort, though -- the simple algorithm seems not to want to converge, at least that was my experience trying to toy around with it. There may be more sophisticated methods out there that could be used.) See here: https://bitbucket.org/bo198214/bunch/downloads/main.pdf for a description of these methods. You'll want to pay attention to the "regular iteration", for it serves as the basis for the Kneser solution, and also because it gives a real-valued extension of tetration for bases in $1 < a \le e^{1/e}$ -- though not one that can be analytically continued to most of the complex plane in terms of bases. Other methods have been proposed which give different answers, however the Kneser solution has a simple uniqueness condition based on the inverse function (see here: http://eretrandre.org/rb/files/Trappmann2009_82.pdf, also see sheldonison's post here: http://math.stackexchange.com/questions/...-tetration -- there's another uniqueness condition given, though I'm not sure if this one has a proof.). Some of these other methods include the "sinh" method, which does not seem to even yield an analytic solution (so cannot be extended to complex heights, and perhaps not even complex bases either), and recently, another method based on divergent summation of "iterate series". The sinh method does appear to be at least smooth, but again, this is not proven. So yes, a lot of methods and proposals, some of which agree with each other, others which don't. And there's a whole lot that's conjecture and not proven. But one can make a case for the Kneser solution due to the simple uniqueness condition, and Robbins' method computes it for at least a limited range of heights and bases in the complex plane. "sheldonison" and I have also written Pari/GP codes that implement the Kneser methods, which give tetration to any complex height, and for possibly (though the current code doesn't allow for it) any complex base: Description of Kneser solution: http://math.eretrandre.org/tetrationforu...hp?tid=213 The algorithm: http://math.eretrandre.org/tetrationforu...hp?tid=486 http://math.eretrandre.org/tetrationforu...hp?tid=487 The code: http://math.eretrandre.org/tetrationforu...hp?tid=486 http://math.eretrandre.org/tetrationforu...hp?tid=729 (sheldonison's) http://math.eretrandre.org/tetrationforu...hp?tid=664 (my own implementation -- you'll want the code from the last message -- I had some difficulties with the first code) http://math.eretrandre.org/tetrationforu...hp?tid=687 (my program for some complex bases) This would, I believe, provide a fair positive answer to "is there a way to extend a^^x to complex a and x". Balarka Sen Junior Fellow Posts: 25 Threads: 7 Joined: Feb 2013 02/19/2013, 11:18 AM I was referring to A. Robbin's paper about extending the tetration to the reals by "the article". My apologies for not being clear to everyone. Thank you, @mike3, your reply helped me a lot. I recently read the kneser's solution to the tetration and saw a code here, in the forum, somewhere. But my problem is that the code seems to be initializing the base and prints all the information on the PARI screen. I want to simultaneously calculate sexp(b, h) for different bases but same heights which is a bit difficult for me using that code, can anyone give me a code that will initialize the height first and can give results for different bases at once? However, I have a question regarding knesers solution to the extended tetration : is it being proved that knesner's method converges for any complex height? I searched a bit (and tried with a pen and paper too ) but haven't found anything regarding it. sheldonison Long Time Fellow Posts: 630 Threads: 22 Joined: Oct 2008 02/19/2013, 05:10 PM (This post was last modified: 02/19/2013, 08:48 PM by sheldonison.) (02/19/2013, 11:18 AM)Balarka Sen Wrote: ... I recently read the kneser's solution to the tetration and saw a code here, in the forum, somewhere. But my problem is that the code seems to be initializing the base and prints all the information on the PARI screen. I want to simultaneously calculate sexp(b, h) for different bases but same heights which is a bit difficult for me using that code, can anyone give me a code that will initialize the height first and can give results for different bases at once? Hi Balarka. I'm Sheldon, the author of the pari-gp code that you probably downloaded. The algorithm has to calculate the equivalent of Kneser's Riemann mapping for each base, before it can generate results, so it won't give you results for different bases as easily as you might like. Quote:However, I have a question regarding knesers solution to the extended tetration : is it being proved that knesner's method converges for any complex height? I searched a bit (and tried with a pen and paper too ) but haven't found anything regarding it. Yes, Kneser's solution for real bases is proven analytic in the upper/lower halves of the complex plane, with singularities at the real axis for negative integers<=-2. I did post a Taylor series for $\text{sexp}_a (-0.5)$ centered at base a=2. Look for "Taylor series for sexp(-0.5)" in the thread with the complex base pari-gp code, http://math.eretrandre.org/tetrationforu...729&page=2 The pari-gp code I wrote that extends tetration to complex bases is much more limited in how well it converges. I regarded the code as an effort to stimulate the discussion as to how to extend tetration to complex bases, based on Mike's ideas. Below is the Taylor series for $^{-0.5} a$, for complex tetration base a centered at a=2. Convergence is limited by the nearest singularity at base $\eta=\exp(\frac{1}{e})\approx1.444667861$, though that is a mild singularity, and using fewer Taylor series terms gives convergence over a wider range, limited more by the singularity at base=1. - Sheldon Code:{ sexp_mhalf = /* sexp(-0.5) for base a-2 */         0.5447641214595567339801218858257244685854 +a^ 1* -0.09026490293475114180982800726025252487179 +a^ 2*  0.05334642698935378617403396491528890594804 +a^ 3* -0.03638190492562309183765608353362070821840 +a^ 4*  0.02665589484943122254265742189263438424835 +a^ 5* -0.02047608577133435850738520805893632252252 +a^ 6*  0.01628939391559684527389871185757624228228 +a^ 7* -0.01331802035638468229849633176805710250959 +a^ 8*  0.01113080347039454404398917618932270486539 +a^ 9* -0.009471945601741301301799666960159500493414 +a^10*  0.008181870472918983418952481797363865140773 +a^11* -0.007156971109633091475785436209879906176635 +a^12*  0.006327698270413005651257016844418549882893 +a^13* -0.005646005057506155565996841622134687059648 +a^14*  0.005077852297548008377590502397935756807242 +a^15* -0.004598579564709383679003395147264261288216 +a^16*  0.004189960720720897899813797031566114828076 +a^17* -0.003838281581489355718364575037825080832826 +a^18*  0.003533055202798846826754080155559369484869 +a^19* -0.003266144873505376098605478730250127897586 +a^20*  0.003031153706186763516973507099991793317656 +a^21* -0.002822992160610217843850530938287773588695 +a^22*  0.002637566583362648226391359841543536218954 +a^23* -0.002471551499838161939220088380143251371568 +a^24*  0.002322220812129558330686955182616565785365 +a^25* -0.002187321052201244713039707812416923885954 +a^26*  0.002064975080105974232286030081964966694861 +a^27* -0.001953608108640999822645409586956590179027 +a^28*  0.001851890298539576237491912671987439471748 +a^29* -0.001758691790434811807511081574014212915353 +a^30*  0.001673047168806168273884584564550319035826 +a^31* -0.001594127148975512837191984637091772279620 +a^32*  0.001521215846034519175841930450931000214014 +a^33* -0.001453692394304569109742037483974061901089 +a^34*  0.001391015984731342378663288972983370893753 +a^35* -0.001332713607717317327426366538502721958881 +a^36*  0.001278369952559800466677607874323180950794 +a^37* -0.001227619037446880124253911053592339307366 +a^38*  0.001180137236855264191223682764928143626889 +a^39* -0.001135637444029102801867724419804092337925 +a^40*  0.001093864160641107689052187995385697434389 +a^41* -0.001054589347847405322195327651270883669674 +a^42*  0.001017608905751088252280461602251417505390 +a^43* -0.0009827396740070429313553656082778187553405 +a^44*  0.0009498168665858747234148639733725614346048 +a^45* -0.0009186918698079533934930814875314534697736 +a^46*  0.0008892303455966866573662840017051026789547 +a^47* -0.0008613105921955727892242590292733293823957 +a^48*  0.0008348221228916550711398042195621478636400 +a^49* -0.0008096644300085595816921514966352297536765 +a^50*  0.0007857459069005338594524734325749675459648 +a^51* -0.0007629829051481069438849922812669102687654 +a^52*  0.0007412989078245228555308404776565708380003 +a^53* -0.0007206238027261275601524070871984237792707 +a^54*  0.0007008932419630012770449395923516096144246 +a^55* -0.0006820480763866325438818551382445843370545 +a^56*  0.0006640338550679383137667663988829019940730 +a^57* -0.0006468003814946490331294868382872885969872 +a^58*  0.0006303013193831178111678644623963748370560 +a^59* -0.0006144938420376038744680011450272121474617 +a^60*  0.0005993383200741946673108758684076009577129 +a^61* -0.0005847980430851115584178527733490186488280 +a^62*  0.0005708389714760195065962019338923580583096 +a^63* -0.0005574295152845303003325133006371836953863 +a^64*  0.0005445403373002463827629604348689289744143 +a^65* -0.0005321441782716469402918017155794455019026 +a^66*  0.0005202157024181592772390603384936993523783 +a^67* -0.0005087313618820156701812678182987572351923 +a^68*  0.0004976692791697413648392248040346423784153 +a^69* -0.0004870091470646722871348673605764087543724 +a^70*  0.0004767321459596961918548061350965022366495 +a^71* -0.0004668208790873150944172435667760750155470 +a^72*  0.0004572593267416065803546985077871002284578 +a^73* -0.0004480328213309398865880156803821028719749 +a^74*  0.0004391280460191844112860195774985849996584 +a^75* -0.0004305330608687577109307488887504560310821 +a^76*  0.0004222373618726151852083171686555860996115 +a^77* -0.0004142319801615488633254732943499557223787 +a^78*  0.0004065096311401751693759190311085411363661 +a^79* -0.0003990649265288728503867805273227150792807 +a^80*  0.0003918946665218884447060007230972331199284 +a^81* -0.0003849982348510417760694970639142865634935 +a^82*  0.0003783781269217428875539830317674205408416 +a^83* -0.0003720406509700054967447945428476416910439 +a^84*  0.0003659968551918890293717641839010994444982 +a^85* -0.0003602637511201750326035146298122448667960 +a^86*  0.0003548659266520138706765657962715630833711 +a^87* -0.0003498376730740476659615080146624803134097 +a^88*  0.0003452257919088022517001222379805220095853 +a^89* -0.0003410933031098378510324434621473247433045 +a^90*  0.0003375243510812609314744505648626776451409 +a^91* -0.0003346307060211903759381304723940593677783 +a^92*  0.0003325603945037557882292885310966766766416 +a^93* -0.0003315091777419398624384857779237117250571 +a^94*  0.0003317358460153907833479388101334190239424 +a^95* -0.0003335826371421790840947235619676003126446 +a^96*  0.0003375025483293688889372157683210232309158 +a^97* -0.0003440959391986930633575807516000693209207 +a^98*  0.0003541596811015852018702302245010936694298 +a^99* -0.0003687532792553722994412436565871821184048 +a^100*  0.0003892879974033369355506985563538084697749 +a^101* -0.0004176472122482091920832394022681079546693 +a^102*  0.0004563492419325118494433069912923916819317 +a^103* -0.0005087680413650084606119114211931555096770 +a^104*  0.0005794328703416784956152178590887291116773 +a^105* -0.0006744359202315312364732927380874961661109 +a^106*  0.0008019877695049482375578463387186199880001 +a^107* -0.0009731755956181394060092634776613016580966 +a^108*  0.001202999930966446829426139160934525077288 +a^109* -0.001511794692123179861160636690582499120690 +a^110*  0.001927175422690043740636959464774028693374 +a^111* -0.002486716638299575521444734874764590083602 +a^112*  0.003241637115621436229181368952656278856890 +a^113* -0.004261880730986434210647558463981630316890 +a^114*  0.005643132413944208166578890706551114196855 +a^115* -0.007516521379651240969156586191270607737992 +a^116*  0.01006206163735266482138789454244816545884 +a^117* -0.01352729756512357159032473435823294438533 +a^118*  0.01825320917728106084640001141407135014463 +a^119* -0.02471025705364750094828467307169149418527 +a^120*  0.03354860917065776303709779278149423244864 +a^121* -0.04566823068915574646795726336973581269876 +a^122*  0.06231683139161434741416351784301382413064 +a^123* -0.08522693583797479716332856332368442457569 +a^124*  0.1168079697188183445926413360961134224671 +a^125* -0.1604158141872473246671531628893015031994 +a^126*  0.2207315839610322539821107999865013181935 +a^127* -0.3042945998433380627917194164914032453560 +a^128*  0.4202533179050196135850984244232386711566 +a^129* -0.5814247303292241247446316577711695736830 +a^130*  0.8057908832077465638002237284863080618377 +a^131* -1.118615564350797221844687119564496999668 +a^132*  1.555441935527913324823119191631251168563 +a^133* -2.166343033000847467475898770333491396571 +a^134*  3.021956187494805723428091061670418388443 +a^135* -4.222060471273729919015929034763285805797 +a^136*  5.907783486640862219163610751023466051097 +a^137* -8.278993788941744624065220674384056537610 +a^138*  11.61911094227013680329706281281624602869 +a^139* -16.33053766929996258807625048927333995682 +a^140*  22.98531974919989325688421392691882638831 +a^141* -32.39766033022431437194340100803250018929 +a^142*  45.72783261862920895891344502067529674014 +a^143* -64.63125037132339003279765074797075722636 +a^144*  91.47255488228885159381088483245314278778 +a^145* -129.6334095048019025160129531044130526933 +a^146*  183.9554899644497666098851981405887199496 +a^147* -261.3787224160197438528735946730321986942 +a^148*  371.8617901668770815153842203841958000090 +a^149* -529.7111364131130428052370084930064042101 +a^150*  755.5016929935324645636711119283331834056 } Balarka Sen Junior Fellow Posts: 25 Threads: 7 Joined: Feb 2013 02/19/2013, 08:07 PM sheldonison Wrote:The algorithm has to calculate the equivalent of Kneser's Riemann mapping for each base, before it can generate results, so it won't give you results for different bases as easily as you might like. I see. So, what about Robbin's method? I think there is a pseudo-code version of his method in his article, on the code section. Can it be implemented in PARI? How much freely can we extend Robin's method to complex numbers? Is there any convergence/divergence issues for Robin's method (complex numbers)? Sorry for asking so many question, actually I am working on a "function" related to the tetration and want to gain full computational access so that I can continue regularizing the function. Is tetration well-defined for negative integers? I mean, what is 2^-3? According to the PARI code you created, it is, in the Kneser sense $7.69052232412789846 + 4.53236014182719380 \cdot i$. Is this a valid argument? Don't we have ${}^{-x} a = \log_{a}(0)$ for positive integer x? Shouldn't we consider that tetration has poles at the negative integers? Balarka . bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 02/20/2013, 12:01 AM (This post was last modified: 02/20/2013, 12:06 AM by bo198214.) (02/18/2013, 11:40 PM)mike3 Wrote: There is also the "continuum sum", the "matrix method" and the above mentioned Robbins' method (which you may be referring to). All five of these appear to yield the same result, but it is not proven, as far as I can tell. In theory, these should be able to define $^z a$ for any complex a and z. But the currently-available implementations only work for more limited ranges of bases a. Ya, Andrew made a picture for the base ranges of the regular tetration (at the attracting fixpoint) and his tetration (which was called "natural tetration" at that time, we changed the nomenclature to "intuitive tetration") in this post. Unfortunately the server doesnt want to be remote linked - and is down anyway - but I could download the picture and add it here in this post.     mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 02/20/2013, 12:06 AM (This post was last modified: 02/20/2013, 12:07 AM by mike3.) (02/20/2013, 12:01 AM)bo198214 Wrote: (02/18/2013, 11:40 PM)mike3 Wrote: There is also the "continuum sum", the "matrix method" and the above mentioned Robbins' method (which you may be referring to). All five of these appear to yield the same result, but it is not proven, as far as I can tell. In theory, these should be able to define $^z a$ for any complex a and z. But the currently-available implementations only work for more limited ranges of bases a. Ya, Andrew made a picture for the base ranges of the regular tetration (at the attracting fixpoint) and his tetration (which was called "natural tetration" at that time, we changed the nomenclature to "intuitive tetration") in this post. Unfortunately the server doesnt want to be crosslinked - and is down anyway - but I could download the picture and add it here in this post. Yes, I forgot about that. Robbins' method does not in fact work for every base. mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 02/20/2013, 12:21 AM (02/19/2013, 08:07 PM)Balarka Sen Wrote: sheldonison Wrote:The algorithm has to calculate the equivalent of Kneser's Riemann mapping for each base, before it can generate results, so it won't give you results for different bases as easily as you might like. I see. So, what about Robbin's method? I think there is a pseudo-code version of his method in his article, on the code section. Can it be implemented in PARI? How much freely can we extend Robin's method to complex numbers? Is there any convergence/divergence issues for Robin's method (complex numbers)? Sorry for asking so many question, actually I am working on a "function" related to the tetration and want to gain full computational access so that I can continue regularizing the function. Is tetration well-defined for negative integers? I mean, what is 2^-3? According to the PARI code you created, it is, in the Kneser sense $7.69052232412789846 + 4.53236014182719380 \cdot i$. Is this a valid argument? Don't we have ${}^{-x} a = \log_{a}(0)$ for positive integer x? Shouldn't we consider that tetration has poles at the negative integers? Balarka . I don't know of a code off-hand, but I suppose I could whip one up. Mmh.... on the question of negative integer heights: 1. On the principal branch, tetration is not defined for negative-integer heights equal to -2 or less. It has branch points (not poles) at those heights. 2. On other branches, there may be a finite value at those points. On these other branches, there will be many additional branch point singularities in the right half-plane. « Next Oldest | Next Newest »

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