(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 , which will converge nicely for z in the range [0..2]. If z<0, take before generating slog(z-1)-1. If z>2, iterate , 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 :/.