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 On the existence of rational operators JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 12/20/2010, 09:38 PM (This post was last modified: 12/20/2010, 09:43 PM by JmsNxn.) (12/20/2010, 08:28 PM)sheldonison Wrote: (12/20/2010, 07:01 PM)JmsNxn Wrote: A taylor series expansion could only work if one also has a slog taylor series expansion. If you give me that I'd be happy to make a graph over domain [0, 2]. Here it is. Taylor series for $\text{slog}_2(z-1)$, which will converge nicely for z in the range [0..2]. If z<0, take $z=2^z$ before generating slog(z-1)-1. If z>2, iterate $z=\log_2(z)$, before generating slog(z-1)+n, so that z is in the range [0..2]. Code:a0=   0.00000000000000000000000000000000 a1=   1.12439780182947880296975296510341 a2=  -0.01233408638319092919966757867732 a3=  -0.15195716580089316798328536602130 a4=   0.01868009944521288546047080416998 a5=   0.03456100685993161409190280063892 a6=  -0.00907417008961111769380973532974 a7=  -0.00882611191544351225979374105298 a8=   0.00382451721437283174576066832193 a9=   0.00228031089800741907723932214202 a10= -0.00151922346582286239408053757523 a11= -0.00055532576725948556607647219741 a12=  0.00058068759568845128571208222568 a13=  0.00011333875202118827889934233768 a14= -0.00021492130432643551427679982642 a15= -0.00001121186618462451210489139936 a16=  0.00007707957627653141354330216317 a17= -0.00000624892419462078938069406186 a18= -0.00002671099173826526447600018191 a19=  0.00000581456717530582001598703419 a20=  0.00000888533730151933862945265998 a21= -0.00000330763773421352876145208923 a22= -0.00000280211888032738989276581338 a23=  0.00000159184385525029832990193555 a24=  0.00000081754010898099012004646318 a25= -0.00000069935182173423145339560199 a26= -0.00000020858066529691830195782405 a27=  0.00000028862436851123339303428296 a28=  0.00000003862884977802212289870391 a29= -0.00000011328157592267567824609526 a30=  0.00000000094217657182114853689258 a31=  0.00000004247233495866309956742421 a32= -0.00000000615478181186900908929094 a33= -0.00000001519770422468823619166244 a34=  0.00000000440788391865597168670175 a35=  0.00000000515674874286180172029316 a36= -0.00000000238020450731772920188547 a37= -0.00000000163450373305911517165825 a38=  0.00000000113080753816867247217337 a39=  0.00000000046806124793717393704457 a40= -0.00000000049617938942328314300887 a41= -0.00000000011074640735137445965583 a42=  0.00000000020520322530692159331762 a43=  0.00000000001419679684872395334262 a44= -0.00000000008069713894112959557995 a45=  0.00000000000566866105603014575315 a46=  0.00000000003024789945770100323766 a47= -0.00000000000635476323844608537350 a48= -0.00000000001077582550414236272297 a49=  0.00000000000393825972677957845029 a50=  0.00000000000361455077455751078451 a51= -0.00000000000202059080878299162503 a52= -0.00000000000111781235812744791900 a53=  0.00000000000093621058575542170278 a54=  0.00000000000030319231965398624547 a55= -0.00000000000040462836021383451852 a56= -0.00000000000006151258021196048097 a57=  0.00000000000016547055506890328371 a58=  0.00000000000000095303168930585396 a59= -0.00000000000006439584830686473714 a60=  0.00000000000000855544017414425242 a61=  0.00000000000002385199982356358626 a62= -0.00000000000000663620464544180588 a63= -0.00000000000000836346245035952446 a64=  0.00000000000000374483888620781880 a65=  0.00000000000000273889585903612029 a66= -0.00000000000000184001327491851237 a67= -0.00000000000000081266892359162900 a68=  0.00000000000000083079153254456575 a69=  0.00000000000000020181213382841263 a70= -0.00000000000000035247402372018293 a71= -0.00000000000000002985852756369760 a72=  0.00000000000000014190787623066308 a73= -0.00000000000000000805476244793420 a74= -0.00000000000000005438469969331962 a75=  0.00000000000000001071535590487842 a76=  0.00000000000000001979662010087425 a77= -0.00000000000000000694989588600789 a78= -0.00000000000000000678704531046374 a79=  0.00000000000000000366768107666540 a80=  0.00000000000000000214963783636005 window screen [xmin = 0, xmax =2, ymin=0, ymax = 10] I'm not liking the way this looks either (don't worry, I used the same algorithm you defined). Are there any other ways to extend tetration that I can try? Hopefully one of them will make the graph without any angles. If not, I guess it just expresses a very odd connection. The function 2 {x} 3 is defined piecewise, so I guess it's just the way it is :/. « Next Oldest | Next Newest »

 Messages In This Thread On the existence of rational operators - by JmsNxn - 12/14/2010, 01:41 AM RE: On the existence of rational operators - by bo198214 - 12/19/2010, 05:23 AM RE: On the existence of rational operators - by tommy1729 - 12/19/2010, 05:21 PM RE: On the existence of rational operators - by JmsNxn - 12/19/2010, 06:47 PM RE: On the existence of rational operators - by sheldonison - 12/19/2010, 08:23 PM RE: On the existence of rational operators - by JmsNxn - 12/20/2010, 02:16 AM RE: On the existence of rational operators - by sheldonison - 12/20/2010, 04:53 AM RE: On the existence of rational operators - by JmsNxn - 12/20/2010, 07:01 PM RE: On the existence of rational operators - by sheldonison - 12/20/2010, 08:28 PM RE: On the existence of rational operators - by JmsNxn - 12/20/2010, 09:38 PM RE: On the existence of rational operators - by bo198214 - 12/20/2010, 09:17 AM RE: On the existence of rational operators - by JmsNxn - 03/11/2011, 07:23 PM RE: On the existence of rational operators - by JmsNxn - 03/19/2011, 05:36 PM

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