Tetration     Gottfried Helms - Univ Kassel        01 - 2008

Continuous Ue-Tetration (f(x) = exp(x)-1, iterated)

Radius of convergence is zero for fractional heights h

In their article "On analytic iteration" Erdös & Jabotinsky [1] stated 1960, that "the function ez-1 was shown by I.N.Baker to have no real non-integer iterates" and cited I.N.Bakers article of 1958.

However, I.N.Baker [2] deals with this question in terms of "radius of convergence", and the focus of his proof is, that "the radius of convergence isnonzero iff the iterator h is integer" (rough translation).

The latter can be shown by numerical examples. However, although the simple inspection seconds this strongly, the zero-radius of convergence does not mean, that to such a powerseries a value could not be assigned principally. If we study powerseries in the well established context of summation of divergent series (see for instance [3] or [4]), we are no more lost in space: we may apply Cesaro-/Euler- or Borel-summation to assign values to a divergent series, if their terms oscillate in sign and "diverge not too strong" (L. Euler).

Thus, if we can show, that the growthrate of the terms of the occuring series "is not too strong", then we may assign values to continuous U-tetration as well, based on those concepts of divergent summation.

From empirical evidence it seems, that the growthrate of terms is roughly hypergeometric (the logarithms of their absolute values increase roughly with their index) and may be Euler- or Borel-summable.

Adding the concept of divergent summation seems to fill the gap, which is the reason for the common expression, that continuous iteration / tetration were not existent/meaningful.

Update 8.12.2010: tables 2 and 3 were wrong; graphical plots added

Table 1) Terms of powerseries for U-tetration 1(x) = exp(x)-1, for different "height" h

The powerseries for h(x) is constructed, if the terms of one column are associated with a power of x, so

1(x) = 0 + 1.0*x + 0.5*x2 + 0.1666*x3 + 0.041666*x4 + ...

 h=1 h=1+1e-10 h=1+1e-9 h=1+1e-8 h=1+1e-7 h=1.000001 h=1.00001 h=1.0001 h=1.001 h=1.01 h=1.1 0 0 0 0 0 0 0 0 0 0 0 1.00000000000 1.00000000000 1.00000000000 1.00000000000 1.00000000000 1.00000000000 1.00000000000 1.00000000000 1.00000000000 1.00000000000 1.00000000000 0.500000000000 0.500000000050 0.500000000500 0.500000005000 0.500000050000 0.500000500000 0.500005000000 0.500050000000 0.500500000000 0.505000000000 0.550000000000 0.166666666667 0.166666666708 0.166666667083 0.166666670833 0.166666708333 0.166667083334 0.166670833358 0.166708335833 0.167083583333 0.170858333333 0.210833333333 0.0416666666667 0.0416666666854 0.0416666668542 0.0416666685417 0.0416666854167 0.0416668541669 0.0416685416938 0.0416854193751 0.0418544376250 0.0435688750000 0.0632500000000 0.00833333333333 0.00833333333903 0.00833333339028 0.00833333390278 0.00833333902778 0.00833339027792 0.00833390279236 0.00833902923627 0.00839042377090 0.00891752145833 0.0156520833333 0.00138888888889 0.00138888889020 0.00138888890197 0.00138888901968 0.00138889019676 0.00138890196764 0.00138901968094 0.00139019726109 0.00140201786294 0.00152479082778 0.00330439236111 0.000198412698413 0.000198412698664 0.000198412700926 0.000198412723545 0.000198412949736 0.000198415211653 0.000198437831961 0.000198664148515 0.000200938693519 0.000224854764284 0.000619448412698 0.0000248015873016 0.0000248015873400 0.0000248015876860 0.0000248015911458 0.0000248016257441 0.0000248019717289 0.0000248054318209 0.0000248400570949 0.0000251887541028 0.0000289291637162 0.000103086328125 0.00000275573192240 0.00000275573192619 0.00000275573196029 0.00000275573230131 0.00000275573571153 0.00000275576981414 0.00000275611087794 0.00000275952529651 0.00000279404961159 0.00000317944735419 0.0000139371209491 0.000000275573192240 0.000000275573193772 0.000000275573207557 0.000000275573345413 0.000000275574723968 0.000000275588509546 0.000000275726367947 0.000000277105214294 0.000000290920211150 0.000000432038819637 0.00000265797691816 0.0000000250521083854 0.0000000250521086406 0.0000000250521109366 0.0000000250521338968 0.0000000250523634998 0.0000000250546595623 0.0000000250776234941 0.0000000253075935284 0.0000000276403744666 0.0000000542870818543 0.000000694792899125 0.00000000208767569879 0.00000000208767508886 0.00000000208766959950 0.00000000208761470595 0.00000000208706577045 0.00000000208157641849 0.00000000202668320538 0.00000000147778177866 -0.00000000400810172166 -0.0000000584935612695 -0.000000504664368756 1.60590438368E-10 1.60590634904E-10 1.60592403728E-10 1.60610091959E-10 1.60786974092E-10 1.62555776881E-10 1.80241951284E-10 0.000000000356918343734 0.00000000210514446846 0.0000000177318932836 -0.00000000421802807312 1.14707455977E-11 1.14711315701E-11 1.14746053210E-11 1.15093428309E-11 1.18567180149E-11 1.53304783117E-11 5.00689269706E-11 0.000000000397537943013 0.00000000388064141333 0.0000000395131940522 0.000000435582178182 7.64716373182E-13 7.64343573715E-13 7.60988378525E-13 7.27436427861E-13 3.91917044815E-13 -2.96326442665E-12 -3.65138432155E-11 -0.000000000371896012790 -0.00000000371333021346 -0.0000000358639150239 -0.000000212854713628 4.77947733239E-14 4.75658969877E-14 4.55060099468E-14 2.49071380137E-14 -1.81081733747E-13 -2.24098569433E-12 -2.28415495699E-11 -2.28999588279E-10 -0.00000000230579288804 -0.0000000245673611746 -0.000000364113506315 2.81145725435E-15 3.37631562397E-15 8.46004094342E-15 5.92972934240E-14 5.67669746844E-13 5.65138714248E-12 5.64878471972E-11 0.000000000564781012687 0.00000000564052431814 0.0000000556347296302 0.000000440314777474 1.56192069686E-16 2.10871750162E-16 7.02988899063E-16 5.62416282482E-15 5.48361457572E-14 5.46980342549E-13 5.47085904392E-12 5.49533060394E-11 0.000000000574130308854 0.00000000818667698740 0.000000302531826895 8.22063524662E-18 -9.08613397676E-16 -9.16011969464E-15 -9.16751827296E-14 -9.16825819610E-13 -9.16833284145E-12 -9.16834682882E-11 -0.000000000916841270162 -0.00000000916898862257 -0.0000000916721406804 -0.000000842960622434 4.11031762331E-19 3.17546572127E-16 3.17176639274E-15 3.17139603745E-14 3.17135477755E-13 3.17130840788E-12 3.17088133269E-11 0.000000000316661410455 0.00000000312392851086 0.0000000269588495460 -0.000000152576230905 1.95729410634E-20 1.68192462253E-15 1.68190700898E-14 1.68190526843E-13 1.68190530239E-12 1.68190738594E-11 1.68192839437E-10 0.00000000168213835683 0.0000000168422404977 0.000000170368161242 0.00000175588878735 8.89679139245E-22 -1.44006757299E-15 -1.44006836535E-14 -1.44006836200E-13 -1.44006753577E-12 -1.44005919420E-11 -1.43997576963E-10 -0.00000000143914142805 -0.0000000143078854015 -0.000000134634795548 -0.000000447507575851 3.86817017063E-23 -3.49777585220E-15 -3.49777589582E-14 -3.49777598736E-13 -3.49777686828E-12 -3.49778567395E-11 -0.000000000349787372745 -0.00000000349875397329 -0.0000000350752749065 -0.000000359233754730 -0.00000412470467839 1.61173757110E-24 5.38854766141E-15 5.38854764161E-14 5.38854745801E-13 5.38854562339E-12 5.38852727729E-11 0.000000000538834381251 0.00000000538650878326 0.0000000536812042036 0.000000518050812288 0.00000301683343116 6.44695028438E-26 8.08723815709E-15 8.08723819079E-14 8.08723852840E-13 8.08724190448E-12 8.08727566529E-11 0.000000000808761326671 0.00000000809098861218 0.0000000812467507604 0.000000845471578850 0.0000109668435898 2.47959626322E-27 -2.08610643846E-14 -2.08610643403E-13 -2.08610638979E-12 -2.08610594741E-11 -2.08610152358E-10 -0.00000000208605728378 -0.0000000208561473527 -0.000000208117422036 -0.00000203528517695 -0.0000145013111830 9.18368986380E-29 -1.98804698507E-14 -1.98804699874E-13 -1.98804713544E-12 -1.98804850238E-11 -1.98806217180E-10 -0.00000000198819886437 -0.0000000198956562364 -0.000000200321652762 -0.00000213800535957 -0.0000326762521589 3.27988923707E-30 8.81620790832E-14 8.81620789730E-13 8.81620778708E-12 8.81620668487E-11 0.000000000881619566273 0.00000000881608543492 0.0000000881498251899 0.000000880388962326 0.00000868663597280 0.0000694414439273 1.13099628864E-31 4.63981159513E-14 4.63981165588E-13 4.63981226334E-12 4.63981833790E-11 0.000000000463987908352 0.00000000464048653567 0.0000000464656064883 0.000000470726072383 0.00000530992672531 0.000106885408600 3.76998762882E-33 -4.13467904020E-13 -4.13467903782E-12 -4.13467901408E-11 -0.000000000413467877659 -0.00000000413467640175 -0.0000000413465265028 -0.000000413441483749 -0.00000413200690203 -0.0000410495847765 -0.000355577741558 1.21612504155E-34 -5.66858222391E-14 -5.66858252371E-13 -5.66858552172E-12 -5.66861550181E-11 -0.000000000566891530263 -0.00000000567191330385 -0.0000000570189261868 -0.000000600161489809 -0.00000899062056549 -0.000369687954408 3.80039075485E-36 2.16041905589E-12 2.16041905589E-11 2.16041905593E-10 0.00000000216041905629 0.0000000216041905989 0.000000216041909431 0.00000216041928365 0.0000216040568546 0.000215872174746 0.00199175357147 1.15163356208E-37 -5.47077213909E-13 -5.47077197440E-12 -5.47077032756E-11 -0.000000000547075385908 -0.00000000547058917425 -0.0000000546894232342 -0.000000545247356215 -0.00000528776126910 -0.0000363878979814 0.00124233837477 3.38715753552E-39 -1.25626433413E-11 -1.25626433480E-10 -0.00000000125626434144 -0.0000000125626440785 -0.000000125626507196 -0.00000125627171216 -0.0000125633802464 -0.000125699219759 -0.00126263688985 -0.0122956957964 9.67759295863E-41 8.40257859633E-12 8.40257849583E-11 0.000000000840257749086 0.00000000840256744117 0.0000000840246694420 0.000000840146196927 0.00000839141170106 0.0000829085750100 0.000728054068510 -0.00291910126219 2.68822026629E-42 8.10158740611E-11 0.000000000810158741453 0.00000000810158749874 0.0000000810158834080 0.000000810159676138 0.00000810168096143 0.0000810252238860 0.000811087929800 0.00818868469497 0.0837613814163 7.26546017915E-44 -9.15286631426E-11 -0.000000000915286624639 -0.00000000915286556770 -0.0000000915285878083 -0.000000915279091204 -0.00000915211221816 -0.0000914532468473 -0.000907739013235 -0.00839237669014 -0.0123753895818 1.91196320504E-45 -0.000000000576761196551 -0.00000000576761197451 -0.0000000576761206446 -0.000000576761296404 -0.00000576762195971 -0.0000576771191244 -0.000576861103416 -0.00577756166342 -0.0586297852900 -0.628707423372 4.90246975651E-47 0.000000000955571716719 0.00000000955571711672 0.0000000955571661193 0.000000955571156410 0.00000955566108576 0.0000955515629597 0.000955010776668 0.00949955952176 0.0898797024579 0.346005170544 1.22561743913E-48 0.00000000450869740944 0.0000000450869741897 0.000000450869751430 0.00000450869846761 0.0000450870800063 0.000450880332773 0.00450975628349 0.0451925428938 0.461104613865 5.18460562367 2.98931082714E-50 -0.0000000102595421705 -0.000000102595421294 -0.00000102595417181 -0.0000102595376059 -0.000102594964837 -0.00102590852542 -0.0102549722775 -0.102137744719 -0.979514574288 -5.10900840368 7.11740673129E-52 -0.0000000384760242903 -0.000000384760243955 -0.00000384760254466 -0.0000384760359579 -0.000384761410707 -0.00384771921724 -0.0384877005134 -0.385925159878 -3.96135098279 -46.8048484100 1.65521086774E-53 0.000000116227861718 0.00000116227861353 0.0000116227857706 0.000116227821235 0.00116227456523 0.0116223809326 0.116187329617 1.15821760352 11.2090207909 68.6891875853 3.76184288123E-55 0.000000356115418772 0.00000356115420001 0.0000356115432287 0.000356115555157 0.00356116783848 0.0356129070510 0.356251912476 3.57477862313 36.9486067074 460.737734874 8.35965084718E-57 -0.00000140454749128 -0.0000140454748778 -0.000140454745280 -0.00140454710305 -0.0140454360555 -0.140450862959 -1.40415877660 -14.0065092062 -136.465277745 -927.516816580 1.81731540156E-58 -0.00000354820288308 -0.0000354820289843 -0.000354820305193 -0.00354820458692 -0.0354821993676 -0.354837343270 -3.54990814732 -35.6523075915 -371.594948853 -4924.66176075 3.86662851396E-60 0.0000181898851298 0.000181898850939 0.00181898847341 0.0181898811366 0.181898451611 1.81894853944 18.1858865205 181.497772157 1777.67618340 13012.1386393 8.05547607075E-62 0.0000377112981275 0.000377112983331 0.00377113003895 0.0377113209530 0.377115265879 3.77135829113 37.7341435490 379.394895942 3996.61935891 56902.5880715 1.64397470832E-63 -0.000252879703867 -0.00252879703474 -0.0252879699546 -0.252879660257 -2.52879267376 -25.2875338389 -252.836031788 -2524.41293675 -24832.8174530 -192427.589592 3.28794941663E-65 -0.000422329790981 -0.00422329793938 -0.0422329823508 -0.422330119212 -4.22333076246 -42.2362646294 -422.658317711 -4256.12115341 -45484.8039105 -707403.008826 6.44695964046E-67 0.00377426676588 0.0377426676139 0.377426671649 3.77426626745 37.7426177707 377.421687072 3773.76755817 37692.4954417 372159.752186 3019868.06295 1.23979993086E-68 0.00489160363873 0.0489160368436 0.489160414068 4.89160870390 48.9165433606 489.211065428 4896.67349905 49422.6804194 539466.261062 9413058.26845 2.33924515256E-70 -0.0604307388101 -0.604307387576 -6.04307382329 -60.4307329855 -604.306805109 -6043.01557255 -60424.9038957 -603719.882403 -5980329.73442 -50445247.9343 4.33193546770E-72 -0.0566594041475 -0.566594049025 -5.66594124522 -56.6594879489 -566.602429160 -5666.77925473 -56743.2849198 -574978.118900 -6500075.49449 -133274607.606 7.87624630492E-74 1.03670175242 10.3670175183 103.670174584 1036.70168597 10367.0108724 103669.509927 1036635.15163 10360288.7816 102928880.951 897923432.867 1.40647255445E-75 0.606849098219 6.06849111592 60.6849245327 606.850582677 6068.63956171 60699.7690680 608334.992559 6217036.42925 75490638.2495 1993335271.96 2.46749570956E-77 -19.0269222427 -190.269222369 -1902.69221781 -19026.9215903 -190269.157127 -1902685.69243 -19026267.7956 -190202521.170 -1894768138.29 -17031649388.3 4.25430294752E-79 -4.42173613236 -44.2173638561 -442.173891809 -4421.76424289 -44220.1749090 -442454.996746 -4449874.38446 -47030827.9756 -723070354.932 -31194996106.3 7.21068296190E-81 372.982633737 3729.82633716 37298.2633506 372982.631407 3729826.10412 37298240.0214 372980273.923 3729565793.59 37247662915.3 344046161128. 1.20178049365E-82 -47.3141710721 -473.141659540 -4731.41147732 -47313.6029646 -473084.848783 -4725730.39960 -46745493.2171 -416272077.263 955549405.335 503699141092. 1.97013195680E-84 -7795.99574192 -77959.9574336 -779599.575779 -7795995.90208 -77959973.4494 -779601176.844 -7796155496.61 -77975420891.4 -780633582303. -7.39446331771E12 3.17763218839E-86 3875.12528713 38751.2517693 387512.407492 3875113.05484 38750028.5403 387390084.361 3862880510.59 37526546856.2 264853251990. -8.19505248641E12 5.04386061649E-88 173446.620895 1734466.20968 17344662.1695 173446628.962 1734467016.36 17344742826.3 173454683290. 1.73526108724E12 1.74127664895E13 1.68897077763E14

Table 2: log10 of absolute values of terms

 h=1 h=1+1e-10 h=1+1e-9 h=1+1e-8 h=1+1e-7 h=1+1e-6 h=1+1e-5 h=1+1e-4 h=1.001 h=1.01 h=1.1 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3006 -0.2967 -0.2596 -0.7782 -0.7782 -0.7782 -0.7782 -0.7782 -0.7782 -0.7781 -0.7780 -0.7771 -0.7674 -0.6761 -1.3802 -1.3802 -1.3802 -1.3802 -1.3802 -1.3802 -1.3802 -1.3800 -1.3783 -1.3608 -1.1989 -2.0792 -2.0792 -2.0792 -2.0792 -2.0792 -2.0792 -2.0792 -2.0789 -2.0762 -2.0498 -1.8054 -2.8573 -2.8573 -2.8573 -2.8573 -2.8573 -2.8573 -2.8573 -2.8569 -2.8532 -2.8168 -2.4809 -3.7024 -3.7024 -3.7024 -3.7024 -3.7024 -3.7024 -3.7024 -3.7019 -3.6969 -3.6481 -3.2080 -4.6055 -4.6055 -4.6055 -4.6055 -4.6055 -4.6055 -4.6055 -4.6048 -4.5988 -4.5387 -3.9868 -5.5598 -5.5598 -5.5598 -5.5598 -5.5598 -5.5598 -5.5597 -5.5592 -5.5538 -5.4976 -4.8558 -6.5598 -6.5598 -6.5598 -6.5598 -6.5598 -6.5597 -6.5595 -6.5574 -6.5362 -6.3645 -5.5754 -7.6012 -7.6012 -7.6012 -7.6012 -7.6012 -7.6011 -7.6007 -7.5967 -7.5585 -7.2653 -6.1581 -8.6803 -8.6803 -8.6803 -8.6803 -8.6805 -8.6816 -8.6932 -8.8304 -8.3971 -7.2329 -6.2970 -9.7943 -9.7943 -9.7943 -9.7942 -9.7937 -9.7890 -9.7441 -9.4474 -8.6767 -7.7512 -8.3749 -10.9404 -10.9404 -10.9403 -10.9389 -10.9260 -10.8144 -10.3004 -9.4006 -8.4111 -7.4033 -6.3609 -12.1165 -12.1167 -12.1186 -12.1382 -12.4068 -11.5282 -10.4375 -9.4296 -8.4302 -7.4453 -6.6719 -13.3206 -13.3227 -13.3419 -13.6037 -12.7421 -11.6496 -10.6413 -9.6402 -8.6372 -7.6096 -6.4388 -14.5511 -14.4716 -14.0726 -13.2270 -12.2459 -11.2478 -10.2480 -9.2481 -8.2487 -7.2547 -6.3562 -15.8063 -15.6760 -15.1531 -14.2499 -13.2609 -12.2620 -11.2619 -10.2600 -9.2410 -8.0869 -6.5192 -17.0851 -15.0416 -14.0381 -13.0377 -12.0377 -11.0377 -10.0377 -9.0377 -8.0377 -7.0378 -6.0742 -18.3861 -15.4982 -14.4987 -13.4987 -12.4988 -11.4988 -10.4988 -9.4994 -8.5053 -7.5693 -6.8165 -19.7083 -14.7742 -13.7742 -12.7742 -11.7742 -10.7742 -9.7742 -8.7741 -7.7736 -6.7686 -5.7555 -21.0508 -14.8416 -13.8416 -12.8416 -11.8416 -10.8416 -9.8416 -8.8419 -7.8444 -6.8708 -6.3492 -22.4125 -14.4562 -13.4562 -12.4562 -11.4562 -10.4562 -9.4562 -8.4561 -7.4550 -6.4446 -5.3846 -23.7927 -14.2685 -13.2685 -12.2685 -11.2685 -10.2685 -9.2685 -8.2687 -7.2702 -6.2856 -5.5204 -25.1906 -14.0922 -13.0922 -12.0922 -11.0922 -10.0922 -9.0922 -8.0920 -7.0902 -6.0729 -4.9599 -26.6056 -13.6807 -12.6807 -11.6807 -10.6807 -9.6807 -8.6807 -7.6808 -6.6817 -5.6914 -4.8386 -28.0370 -13.7016 -12.7016 -11.7016 -10.7016 -9.7016 -8.7015 -7.7012 -6.6983 -5.6700 -4.4858 -29.4841 -13.0547 -12.0547 -11.0547 -10.0547 -9.0547 -8.0547 -7.0548 -6.0553 -5.0611 -4.1584 -30.9465 -13.3335 -12.3335 -11.3335 -10.3335 -9.3335 -8.3334 -7.3329 -6.3272 -5.2749 -3.9711 -32.4237 -12.3836 -11.3836 -10.3836 -9.3836 -8.3836 -7.3836 -6.3836 -5.3838 -4.3867 -3.4491 -33.9150 -13.2465 -12.2465 -11.2465 -10.2465 -9.2465 -8.2463 -7.2440 -6.2217 -5.0462 -3.4322 -35.4202 -11.6655 -10.6655 -9.6655 -8.6655 -7.6655 -6.6655 -5.6655 -4.6655 -3.6658 -2.7008 -36.9387 -12.2620 -11.2620 -10.2620 -9.2620 -8.2620 -7.2621 -6.2634 -5.2767 -4.4390 -2.9058 -38.4702 -10.9009 -9.9009 -8.9009 -7.9009 -6.9009 -5.9009 -4.9009 -3.9007 -2.8987 -1.9102 -40.0142 -11.0756 -10.0756 -9.0756 -8.0756 -7.0756 -6.0756 -5.0762 -4.0814 -3.1378 -2.5348 -41.5705 -10.0914 -9.0914 -8.0914 -7.0914 -6.0914 -5.0914 -4.0914 -3.0909 -2.0868 -1.0770 -43.1387 -10.0384 -9.0384 -8.0384 -7.0384 -6.0384 -5.0385 -4.0388 -3.0420 -2.0761 -1.9074 -44.7185 -9.2390 -8.2390 -7.2390 -6.2390 -5.2390 -4.2390 -3.2389 -2.2383 -1.2319 -0.2016 -46.3096 -9.0197 -8.0197 -7.0197 -6.0197 -5.0197 -4.0198 -3.0200 -2.0223 -1.0463 -0.4609 -47.9116 -8.3459 -7.3459 -6.3459 -5.3459 -4.3459 -3.3459 -2.3458 -1.3449 -0.3362 0.7147 -49.5244 -7.9889 -6.9889 -5.9889 -4.9889 -3.9889 -2.9889 -1.9891 -0.9908 -0.0090 0.7083 -51.1477 -7.4148 -6.4148 -5.4148 -4.4148 -3.4148 -2.4148 -1.4147 -0.4135 0.5978 1.6703 -52.7811 -6.9347 -5.9347 -4.9347 -3.9347 -2.9347 -1.9347 -0.9348 0.0638 1.0496 1.8369 -54.4246 -6.4484 -5.4484 -4.4484 -3.4484 -2.4484 -1.4484 -0.4482 0.5532 1.5676 2.6635 -56.0778 -5.8525 -4.8525 -3.8525 -2.8525 -1.8525 -0.8525 0.1474 1.1463 2.1350 2.9673 -57.7406 -5.4500 -4.4500 -3.4500 -2.4500 -1.4500 -0.4500 0.5502 1.5521 2.5701 3.6924 -59.4127 -4.7402 -3.7402 -2.7402 -1.7402 -0.7402 0.2598 1.2597 2.2589 3.2499 4.1143 -61.0939 -4.4235 -3.4235 -2.4235 -1.4235 -0.4235 0.5765 1.5767 2.5791 3.6017 4.7551 -62.7841 -3.5971 -2.5971 -1.5971 -0.5971 0.4029 1.4029 2.4028 3.4022 4.3950 5.2843 -64.4831 -3.3743 -2.3743 -1.3743 -0.3743 0.6257 1.6257 2.6260 3.6290 4.6579 5.8497 -66.1906 -2.4232 -1.4232 -0.4232 0.5768 1.5768 2.5768 3.5768 4.5763 5.5707 6.4800 -67.9066 -2.3105 -1.3105 -0.3105 0.6895 1.6895 2.6895 3.6899 4.6939 5.7320 6.9737 -69.6309 -1.2187 -0.2187 0.7813 1.7813 2.7813 3.7813 4.7812 5.7808 6.7767 7.7028 -71.3633 -1.2467 -0.2467 0.7533 1.7533 2.7533 3.7533 4.7539 5.7597 6.8129 8.1247 -73.1037 0.0157 1.0157 2.0157 3.0157 4.0157 5.0157 6.0156 7.0154 8.0125 8.9532 -74.8519 -0.2169 0.7831 1.7831 2.7831 3.7831 4.7832 5.7841 6.7936 7.8779 9.2996 -76.6077 1.2794 2.2794 3.2794 4.2794 5.2794 6.2794 7.2794 8.2792 9.2776 10.2313 -78.3712 0.6456 1.6456 2.6456 3.6456 4.6456 5.6459 6.6483 7.6724 8.8592 10.4941 -80.1420 2.5717 3.5717 4.5717 5.5717 6.5717 7.5717 8.5717 9.5717 10.5711 11.5366 -81.9202 1.6750 2.6750 3.6750 4.6750 5.6749 6.6745 7.6697 8.6194 8.9803 11.7022 -83.7055 3.8919 4.8919 5.8919 6.8919 7.8919 8.8919 9.8919 10.8920 11.8924 12.8689 -85.4979 3.5883 4.5883 5.5883 6.5883 7.5883 8.5881 9.5869 10.5743 11.4230 12.9136 -87.2972 5.2392 6.2392 7.2392 8.2392 9.2392 10.2392 11.2392 12.2394 13.2409 14.2276

The precise growthrate of terms is not yet determined. However, it seems, that the forward differences (along columns) for all documented columns tend to be roughly equal for the fractional heights, perhaps depending on some index-shift.

Table 3: forward differences 1'st order of log10 of absolute values of terms

 h=1 h=1+1e-10 h=1+1e-9 h=1+1e-8 h=1+1e-7 h=1+1e-6 h=1+1e-5 h=1+1e-4 h=1.001 h=1.01 h=1.1 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3010 -0.3006 -0.2967 -0.2596 -0.4771 -0.4771 -0.4771 -0.4771 -0.4771 -0.4771 -0.4771 -0.4771 -0.4765 -0.4707 -0.4164 -0.6021 -0.6021 -0.6021 -0.6021 -0.6021 -0.6021 -0.6021 -0.6020 -0.6012 -0.5935 -0.5229 -0.6990 -0.6990 -0.6990 -0.6990 -0.6990 -0.6990 -0.6990 -0.6989 -0.6980 -0.6889 -0.6065 -0.7782 -0.7782 -0.7782 -0.7782 -0.7782 -0.7782 -0.7781 -0.7780 -0.7770 -0.7670 -0.6755 -0.8451 -0.8451 -0.8451 -0.8451 -0.8451 -0.8451 -0.8451 -0.8450 -0.8437 -0.8313 -0.7271 -0.9031 -0.9031 -0.9031 -0.9031 -0.9031 -0.9031 -0.9031 -0.9030 -0.9019 -0.8906 -0.7788 -0.9542 -0.9542 -0.9542 -0.9542 -0.9542 -0.9542 -0.9543 -0.9543 -0.9550 -0.9590 -0.8690 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -0.9998 -0.9982 -0.9825 -0.8668 -0.7196 -1.0414 -1.0414 -1.0414 -1.0414 -1.0414 -1.0414 -1.0412 -1.0394 -1.0222 -0.9008 -0.5827 -1.0792 -1.0792 -1.0792 -1.0792 -1.0793 -1.0805 -1.0925 -1.2336 -0.8386 0.0324 -0.1389 -1.1139 -1.1139 -1.1139 -1.1139 -1.1133 -1.1074 -1.0509 -0.6170 -0.2797 -0.5184 -2.0779 -1.1461 -1.1461 -1.1460 -1.1447 -1.1323 -1.0254 -0.5563 0.0468 0.2656 0.3480 2.0140 -1.1761 -1.1763 -1.1784 -1.1993 -1.4808 -0.7138 -0.1371 -0.0290 -0.0191 -0.0421 -0.3110 -1.2041 -1.2060 -1.2233 -1.4655 -0.3353 -0.1213 -0.2037 -0.2106 -0.2069 -0.1643 0.2332 -1.2304 -1.1489 -0.7307 0.3767 0.4962 0.4017 0.3932 0.3920 0.3885 0.3550 0.0825 -1.2553 -1.2044 -1.0804 -1.0230 -1.0150 -1.0142 -1.0139 -1.0119 -0.9923 -0.8322 -0.1630 -1.2788 0.6344 1.1150 1.2122 1.2232 1.2243 1.2242 1.2223 1.2033 1.0491 0.4450 -1.3010 -0.4566 -0.4606 -0.4610 -0.4610 -0.4611 -0.4611 -0.4617 -0.4676 -0.5315 -0.7423 -1.3222 0.7240 0.7245 0.7246 0.7246 0.7246 0.7246 0.7253 0.7317 0.8007 1.0610 -1.3424 -0.0674 -0.0674 -0.0674 -0.0674 -0.0674 -0.0675 -0.0678 -0.0708 -0.1022 -0.5937 -1.3617 0.3854 0.3854 0.3854 0.3854 0.3854 0.3854 0.3858 0.3894 0.4262 0.9646 -1.3802 0.1877 0.1877 0.1877 0.1877 0.1877 0.1877 0.1874 0.1848 0.1590 -0.1358 -1.3979 0.1763 0.1763 0.1763 0.1763 0.1763 0.1764 0.1767 0.1800 0.2127 0.5605 -1.4150 0.4115 0.4115 0.4115 0.4115 0.4115 0.4115 0.4112 0.4085 0.3815 0.1213 -1.4314 -0.0209 -0.0209 -0.0209 -0.0209 -0.0209 -0.0209 -0.0205 -0.0166 0.0214 0.3528 -1.4472 0.6469 0.6469 0.6469 0.6469 0.6469 0.6468 0.6465 0.6429 0.6088 0.3274 -1.4624 -0.2788 -0.2788 -0.2788 -0.2788 -0.2788 -0.2787 -0.2781 -0.2719 -0.2138 0.1873 -1.4771 0.9499 0.9499 0.9499 0.9499 0.9499 0.9499 0.9493 0.9434 0.8882 0.5220 -1.4914 -0.8630 -0.8630 -0.8630 -0.8630 -0.8629 -0.8627 -0.8604 -0.8379 -0.6595 0.0169 -1.5051 1.5811 1.5811 1.5811 1.5811 1.5810 1.5808 1.5785 1.5563 1.3804 0.7314 -1.5185 -0.5965 -0.5965 -0.5965 -0.5965 -0.5965 -0.5966 -0.5979 -0.6113 -0.7732 -0.2050 -1.5315 1.3610 1.3610 1.3610 1.3610 1.3610 1.3612 1.3625 1.3761 1.5403 0.9955 -1.5441 -0.1747 -0.1747 -0.1747 -0.1747 -0.1747 -0.1747 -0.1753 -0.1807 -0.2391 -0.6245 -1.5563 0.9842 0.9842 0.9842 0.9842 0.9842 0.9842 0.9848 0.9905 1.0511 1.4578 -1.5682 0.0530 0.0530 0.0530 0.0530 0.0530 0.0529 0.0526 0.0489 0.0107 -0.8305 -1.5798 0.7994 0.7994 0.7994 0.7994 0.7994 0.7995 0.7999 0.8038 0.8442 1.7059 -1.5911 0.2193 0.2193 0.2193 0.2193 0.2193 0.2192 0.2189 0.2160 0.1855 -0.2594 -1.6021 0.6738 0.6738 0.6738 0.6738 0.6738 0.6738 0.6741 0.6774 0.7101 1.1756 -1.6128 0.3571 0.3571 0.3571 0.3571 0.3571 0.3570 0.3568 0.3541 0.3272 -0.0064 -1.6232 0.5741 0.5741 0.5741 0.5741 0.5741 0.5741 0.5744 0.5773 0.6068 0.9620 -1.6335 0.4801 0.4801 0.4801 0.4801 0.4801 0.4801 0.4798 0.4773 0.4517 0.1666 -1.6435 0.4863 0.4863 0.4863 0.4863 0.4863 0.4863 0.4866 0.4895 0.5180 0.8266 -1.6532 0.5959 0.5959 0.5959 0.5959 0.5959 0.5959 0.5957 0.5931 0.5674 0.3039 -1.6628 0.4025 0.4025 0.4025 0.4025 0.4025 0.4025 0.4028 0.4058 0.4350 0.7251 -1.6721 0.7098 0.7098 0.7098 0.7098 0.7098 0.7098 0.7095 0.7068 0.6798 0.4220 -1.6812 0.3166 0.3166 0.3166 0.3166 0.3166 0.3167 0.3170 0.3202 0.3518 0.6408 -1.6902 0.8264 0.8264 0.8264 0.8264 0.8264 0.8264 0.8261 0.8231 0.7933 0.5291 -1.6990 0.2227 0.2227 0.2227 0.2227 0.2227 0.2228 0.2232 0.2269 0.2628 0.5654 -1.7076 0.9512 0.9512 0.9512 0.9512 0.9512 0.9511 0.9508 0.9472 0.9129 0.6303 -1.7160 0.1126 0.1126 0.1126 0.1126 0.1126 0.1127 0.1131 0.1177 0.1612 0.4937 -1.7243 1.0918 1.0918 1.0918 1.0918 1.0918 1.0918 1.0913 1.0869 1.0448 0.7291 -1.7324 -0.0280 -0.0280 -0.0280 -0.0280 -0.0280 -0.0279 -0.0273 -0.0212 0.0362 0.4219 -1.7404 1.2624 1.2624 1.2624 1.2624 1.2624 1.2623 1.2617 1.2557 1.1996 0.8285 -1.7482 -0.2326 -0.2326 -0.2326 -0.2326 -0.2326 -0.2325 -0.2315 -0.2218 -0.1346 0.3463 -1.7559 1.4963 1.4963 1.4963 1.4963 1.4963 1.4962 1.4952 1.4856 1.3997 0.9317 -1.7634 -0.6338 -0.6338 -0.6338 -0.6338 -0.6337 -0.6335 -0.6310 -0.6068 -0.4184 0.2628 -1.7709 1.9261 1.9261 1.9261 1.9261 1.9261 1.9258 1.9233 1.8993 1.7119 1.0425 -1.7782 -0.8967 -0.8967 -0.8967 -0.8967 -0.8967 -0.8972 -0.9019 -0.9523 -1.5908 0.1656 -1.7853 2.2169 2.2169 2.2169 2.2169 2.2169 2.2174 2.2221 2.2726 2.9122 1.1667 -1.7924 -0.3036 -0.3036 -0.3036 -0.3036 -0.3036 -0.3037 -0.3050 -0.3176 -0.4694 0.0446 -1.7993 1.6509 1.6509 1.6509 1.6509 1.6509 1.6510 1.6523 1.6650 1.8179 1.3141

It is interesting to see the ratio of consecutive coefficients graphically.

Formal powerseries, whose consecutive coefficients approach a constant ratio have a finite radius of convergence. Formal powerseries, whose radius of convergence is infinite must have a hypergeometric rate of decrease, as for instance the formal powerseries for exp(x). Then there are formal powerseries whose ratio increases with the index, for instance the Eulerian powerseries 1!x  2!x^2+3!x^3- Such formal powerseries have convergence-radius zero.

As I.N. Baker has shown, all fractional iterates of f(x)=exp(x)-1 have convergence-radius zero. But they show an interesting pattern of the increase of the ratio of concecutive oefficients. In the following plot I show the log10 of the ratios (actually of the absoulute value) in a double-logarithmic scale.

The blue line represents the f°1(x) at the integer height 1; the magenta line that of f°1+µ(x), with µ=10^-10 which means a very near-integer-iterate.

The blue line decreases linearly with the index k, which reflects, that the ratio r(k) of two coefficients of the powerseries for exp(x)-1 at the index k is just (k-1)!/k! = 1/k and thus the represented function is entire. The same ratios in the fractional iterate follow that of the integer iterate, but then begin to deviate and show oscillating behaviour. Additionally, that oscillation is also overlaid by an increase which seems to be linear in this scale.

If we look at a fractional iterate farther away from the integer-iterate we get the next plot, where now the green line indicates f°1+µ(x) where µ=10^-1. We get the same pattern, only that the point of deviation and the beginning of oscillation is at an earlier index.

This suggests another, surprising, interpretation of the exponential-series itself: the powerseries of integer-iterates are only limits, where the index of beginning of increase is shifted to infinity  and thus does no more appear

For a computation of f°0.5(x) see the example at [6], where I computed the series using Euler-summation of order 2.5.

If f(x) = exp(x) 1 is generalized to arbitrary bases fb(x) = bx-1, the terms of the powerseries for exp(x)-1 must be scaled by powers of u=log(b). If u<>1 (or is left symbolically) an eigensystem-decomposition of the associated triangular matrix Ub can be performed and the diagonal (and eigenvalues) are the powers of u. An arbitrary height h of iteration can then be determined by substituting u by the h'th power of u in the set of eigenvalues. Then the fractional power of Ubh can be symbolically described and in its second column we have the coefficients for the associated powerseries for the continuous iterated fb°h(x).

Here are the first four terms of the powerseries of height h computed via the symbolic eigensystem-decomposition of the U-tetration-matrix S2b (see other articles in tetration-index [5])

fb°h (x)         = uh                * x /1!
+ uh u (uh-1)/(u-1)                *x2 /2!
+ uh u2(uh-1)/(u-1)* [(uh-1)(u+1)+(uh-u)]/(u2-1)   * x3/3!
+ ...

We see, that, if u=1 , which is the case when fb(x) = exp(x)-1, the coefficients at x have zeros in the denominator, and produce a singularity, if not uh is a positive integer power of u and numerators cancel the denominators. By shift of the parameter x this is convertible to the usual-tetration with base eta=e^(1/e) and answers then the same question for this version of tetration.

Thus the eigensystem-decomposition may be a good approach to go to study the general question of integer vs fractional iteration in more detail.

Gottfried Helms, 5.1.2008

## References

[1] Erdös, Paul , Jabotinsky, Eri;  On analytic iteration , J. Anal. Math. 8, 361-376 (1961)  (also online at digicenter göttingen)

[2] Baker, I.N.; Zusammensetzung ganzer Funktionen, Math Zeitschr. Bd. 69 pp 121-163 (1958)  (also online at digicenter göttingen)

[3] Knopp, Konrad; Theorie und Anwendung unendlicher Reihen, Springer, 1964  (also online at digicenter göttingen)

[4] Hardy, G. H.; Divergent Series; New York: Oxford University Press, 1949

[5] Helms:Tetration-Index

[6] Helms:U-tetration, h=1/2       http://go.helms-net.de/math/tetdocs/CoefficientsForUTetration.htm

Snippets

[1] Erdös/Jabotinsky Pg 362

[1] Erdös/Jabotinsky Pg 376

[2] Baker Pg 161

Gottfried Helms,  8.12.2010, first version: 05.01.2008