Bummer... did a more accurate measurement on the convergence -- looks like 100 coefficients in each direction will only give 15-16 digits of accuracy.
Here's the coefficients for the tetrational base \( e \), modulated by the Gaussian as shown:
(The small imaginary part is a round off error):
and the negative-degree coefficients:
Here's the coefficients for the tetrational base \( e \), modulated by the Gaussian as shown:
(The small imaginary part is a round off error):
Code:
a[0] = 0.397875391058100595418111 + 2.9850441245110140462 E-61*I
a[1] = 0.416189154885943649018174 + 1.9195842352032945454 E-60*I
a[2] = 0.145193802913168035746894 - 6.153556682827798856 E-61*I
a[3] = -0.0329201654231789117932072 - 6.609222044925913430 E-61*I
a[4] = -0.0146936116956140366967053 - 1.0607130827070990655 E-60*I
a[5] = 0.0110622948810356852131289 + 0.E-61*I
a[6] = -0.00172833502092149930384961 + 3.994678610137473554 E-61*I
a[7] = -0.00180835508937351785635017 + 1.2661662572059834638 E-60*I
a[8] = 0.00157515772698227975979782 - 6.186108626681812204 E-61*I
a[9] = -0.000535236409811992349076167 + 8.966021985857690207 E-61*I
a[10] = -9.77675994121522083519057 E-5 + 1.3849254092325444506 E-60*I
a[11] = 0.000250010664600388723240219 - 9.227946283399708211 E-61*I
a[12] = -0.000170812756598325659089718 + 0.E-61*I
a[13] = 5.73706487998321205152364 E-5 + 0.E-61*I
a[14] = 1.16033585501542046248279 E-5 - 5.400813015342964984 E-62*I
a[15] = -3.22628720257092790720677 E-5 + 6.273421451219000834 E-61*I
a[16] = 2.59666125133106224062854 E-5 - 4.462573445252261917 E-61*I
a[17] = -1.23572147925053013371470 E-5 - 1.4900556311994949756 E-60*I
a[18] = 1.71404012816780638286111 E-6 + 0.E-61*I
a[19] = 3.39074764592592092593282 E-6 + 5.546244708549837783 E-61*I
a[20] = -4.23330953435767752463413 E-6 - 1.4593180859561573286 E-60*I
a[21] = 2.99761918164106298894254 E-6 + 5.859022626312693552 E-61*I
a[22] = -1.35468891893702642331490 E-6 + 2.0183108005574934644 E-60*I
a[23] = 1.38465637192648226018902 E-7 + 2.0525080575792498574 E-60*I
a[24] = 4.59321733662540982796999 E-7 - 9.606564334553191880 E-61*I
a[25] = -5.76687416451818575771329 E-7 - 1.1160783086457494800 E-60*I
a[26] = 4.39224799220836982766874 E-7 - 1.4513595207362045963 E-60*I
a[27] = -2.31697536804857357892810 E-7 + 4.134703860459057680 E-61*I
a[28] = 5.92620019777023837219732 E-8 + 6.385664986253645420 E-61*I
a[29] = 4.20898986292613309771500 E-8 - 1.2450292354861060336 E-60*I
a[30] = -7.84364193301053660743543 E-8 + 4.587242661830708352 E-61*I
a[31] = 7.27398375979498645162490 E-8 + 3.782960000987940630 E-61*I
a[32] = -4.85109609270205489457946 E-8 + 7.224531536018421310 E-61*I
a[33] = 2.23861456942567905105987 E-8 + 2.2410168612528228352 E-60*I
a[34] = -2.86459520282896688975806 E-9 + 5.425108648749196066 E-61*I
a[35] = -7.85379007070825266641794 E-9 - 1.6805656882674358601 E-60*I
a[36] = 1.11868036739744560753676 E-8 - 1.5987419640240867624 E-61*I
a[37] = -9.88486022637174088592919 E-9 + 1.2657073222455788301 E-60*I
a[38] = 6.57973079390423678632267 E-9 + 1.0896547868791202914 E-61*I
a[39] = -3.12685133933201448329449 E-9 + 1.2245179540360111058 E-60*I
a[40] = 5.00793151317478543886855 E-10 - 1.0847905575555328089 E-60*I
a[41] = 1.01921311820734388923147 E-9 - 1.9705505216136802586 E-61*I
a[42] = -1.57486101968126885772701 E-9 + 0.E-61*I
a[43] = 1.48382725528176917888905 E-9 + 5.800591992630454416 E-61*I
a[44] = -1.07278266135766583611448 E-9 + 2.3673656365209571122 E-60*I
a[45] = 5.91393342126508110476373 E-10 + 2.2624948620853793066 E-60*I
a[46] = -1.88086564992327356679738 E-10 + 0.E-61*I
a[47] = -7.73625553845716638140121 E-11 - 5.988289000338183422 E-61*I
a[48] = 2.05846212745759269838244 E-10 - 1.4353069532990601708 E-60*I
a[49] = -2.29938243046527653804108 E-10 + 0.E-61*I
a[50] = 1.91340971688285908262136 E-10 + 1.0000753744597080188 E-60*I
a[51] = -1.26965919854134121032240 E-10 + 3.354005847534243129 E-61*I
a[52] = 6.27241433951610870046252 E-11 + 2.6680656443800172726 E-61*I
a[53] = -1.27554698490738113179862 E-11 + 3.651119749002290437 E-61*I
a[54] = -1.83236230992450431858269 E-11 + 0.E-61*I
a[55] = 3.20921606979202761627920 E-11 + 2.5966855895604019986 E-61*I
a[56] = -3.31681014233249216857662 E-11 - 9.166436613722241994 E-61*I
a[57] = 2.68392089286889387832932 E-11 + 1.7914157461000540476 E-60*I
a[58] = -1.76318602279710468751399 E-11 + 1.3166943389636563108 E-60*I
a[59] = 8.69284889345558056935594 E-12 + 6.080470517003548930 E-61*I
a[60] = -1.74232704665226971623959 E-12 - 2.4339713208887516650 E-61*I
a[61] = -2.65334914006790198704635 E-12 - 1.6245329612991390394 E-60*I
a[62] = 4.69145775573836397670701 E-12 - 1.5498501075859591586 E-60*I
a[63] = -4.95935435400045178020460 E-12 - 8.557056786077598276 E-61*I
a[64] = 4.14672723475903838162245 E-12 + 1.1072489386005317462 E-60*I
a[65] = -2.86377465983331342292356 E-12 + 1.4859188838587007500 E-60*I
a[66] = 1.55478646551640234218994 E-12 + 0.E-61*I
a[67] = -4.82139005441705749124474 E-13 - 8.889365529047513239 E-61*I
a[68] = -2.47143106719271718406506 E-13 - 9.196594103143828457 E-61*I
a[69] = 6.36114497738363193531291 E-13 - 2.1486380323392822628 E-61*I
a[70] = -7.50874402464770287965930 E-13 + 1.6320674706080463846 E-60*I
a[71] = 6.81586835289400046205588 E-13 + 1.6223001498193322858 E-60*I
a[72] = -5.15514528195414641348527 E-13 + 2.1512095618115911032 E-60*I
a[73] = 3.22004369448179887030840 E-13 + 2.0167120121818167768 E-60*I
a[74] = -1.47127859276329591059006 E-13 + 1.3530384902365274826 E-60*I
a[75] = 1.48619138606160662654643 E-14 - 2.2107123486648895330 E-60*I
a[76] = 6.81806683696976514667217 E-14 - 1.5956928302738256176 E-60*I
a[77] = -1.06835153965874312282071 E-13 - 1.8502631798810804800 E-60*I
a[78] = 1.11855715496354125502501 E-13 - 2.3301484139823512510 E-61*I
a[79] = -9.56074174059497298207089 E-14 + 3.459308371959962772 E-60*I
a[80] = 6.92406912140772493555385 E-14 + 2.0177144019872742998 E-60*I
a[81] = -4.12344666488088741457145 E-14 + 1.4027704083224675329 E-60*I
a[82] = 1.69989528696582883423590 E-14 - 1.1939093710245961698 E-60*I
a[83] = 8.28557333333947259946431 E-16 - 1.2812244721545093977 E-60*I
a[84] = -1.17301959482471538616070 E-14 + 0.E-61*I
a[85] = 1.65548785804244879058777 E-14 + 9.250786505007335062 E-61*I
a[86] = -1.68354953604888918779494 E-14 + 1.6947032525180505029 E-60*I
a[87] = 1.42635476691793834697834 E-14 + 4.586704609864632169 E-61*I
a[88] = -1.03462501346013446634608 E-14 - 5.956702009493176365 E-61*I
a[89] = 6.23014521174691018831580 E-15 - 1.8923225574097490690 E-60*I
a[90] = -2.65386727870540686082214 E-15 - 9.457280254834658315 E-61*I
a[91] = -1.36743824456341755390483 E-17 - 9.198759367435836521 E-61*I
a[92] = 1.68991171106908868158372 E-15 + 0.E-61*I
a[93] = -2.48151945854288445675137 E-15 + 2.559961278697615104 E-60*I
a[94] = 2.59472725640198465329043 E-15 + 1.2416813791761960441 E-60*I
a[95] = -2.26454587544363434925095 E-15 + 8.848369631381160165 E-61*I
a[96] = 1.70679875149646918218844 E-15 - 7.455253132690566890 E-61*I
a[97] = -1.09166406110655826382257 E-15 - 1.2893324994708104573 E-60*I
a[98] = 5.34526532821100880705770 E-16 + 0.E-61*I
a[99] = -9.88572418981299229968335 E-17 + 0.E-61*I
a[100] = -1.93977952794815338960238 E-16 - 1.1932106382924937024 E-60*Iand the negative-degree coefficients:
Code:
a[-1] = 0.118714366068275491139864 + 1.2835397909662120082 E-60*I
a[-2] = -0.0369421953926547148001681 - 9.773327445418275734 E-61*I
a[-3] = -0.00955787685160441768978072 - 2.3683477764127354318 E-60*I
a[-4] = 0.0105346633896152201891680 - 4.750906105176736143 E-61*I
a[-5] = -0.00278544093753364182911849 - 6.812575863861415539 E-61*I
a[-6] = -0.00113746278912202521777083 + 1.3525989300396415025 E-60*I
a[-7] = 0.00149606845332274563487211 + 5.187478321895119788 E-61*I
a[-8] = -0.000692053329891718522941987 + 5.685336628751597375 E-61*I
a[-9] = 3.72699540037986232050485 E-5 - 1.6031405792330813983 E-60*I
a[-10] = 0.000202673430359728381444716 + 0.E-61*I
a[-11] = -0.000181155346083026835684956 + 1.5710306168750673432 E-61*I
a[-12] = 8.31323708325276189154988 E-5 + 0.E-61*I
a[-13] = -6.60938888491505880235531 E-6 - 4.790275268171344574 E-61*I
a[-14] = -2.60140940617459515381288 E-5 - 1.1626175538036606982 E-60*I
a[-15] = 2.74318412498564493421101 E-5 - 7.947392844057401237 E-61*I
a[-16] = -1.62573052848227757666247 E-5 - 7.764172327980947514 E-61*I
a[-17] = 4.87293045103624148056718 E-6 + 1.3446294372037274322 E-60*I
a[-18] = 1.91116681343650782771301 E-6 - 4.089456297645928344 E-61*I
a[-19] = -4.09267008579672159051381 E-6 + 4.741043558227001726 E-61*I
a[-20] = 3.49558406867503292406780 E-6 + 2.0642426247743434862 E-60*I
a[-21] = -1.93628188950229441011984 E-6 + 1.8313309409859436603 E-60*I
a[-22] = 5.32401910689934060147273 E-7 + 0.E-61*I
a[-23] = 2.99062098588705940667515 E-7 - 9.676901700219074217 E-61*I
a[-24] = -5.82707454307740529747510 E-7 - 1.0218183086733718353 E-60*I
a[-25] = 5.20338395105059715584950 E-7 + 4.753551074903132911 E-61*I
a[-26] = -3.19579982184098489760597 E-7 - 1.4905070317308039177 E-61*I
a[-27] = 1.20122734827712474542618 E-7 + 3.0131320758905032910 E-61*I
a[-28] = 1.47054770875714223143751 E-8 + 0.E-61*I
a[-29] = -7.63225621367399383010778 E-8 - 5.968455892835055058 E-61*I
a[-30] = 8.36791727655117218101898 E-8 + 0.E-61*I
a[-31] = -6.24246309431715664962425 E-8 + 1.1255404619245750633 E-60*I
a[-32] = 3.34462516932598301110728 E-8 - 2.6583464337186196360 E-61*I
a[-33] = -9.10688232594550776584457 E-9 + 9.540970786115621652 E-61*I
a[-34] = -5.92779372572677067492525 E-9 - 1.9650874852508057502 E-61*I
a[-35] = 1.20323850083452019291928 E-8 - 6.483548289837628729 E-61*I
a[-36] = -1.19093229700253504587441 E-8 + 4.035975234800842194 E-61*I
a[-37] = 8.63559964962720920997923 E-9 - 1.3145152429715187371 E-60*I
a[-38] = -4.62051640065370451908076 E-9 + 0.E-61*I
a[-39] = 1.28561146893290583816355 E-9 - 5.283791543417498821 E-61*I
a[-40] = 8.22863985770012259991888 E-10 - 1.0457773727539229825 E-60*I
a[-41] = -1.74394101069019941430219 E-9 + 2.2447070902738456334 E-60*I
a[-42] = 1.80338785172728153254588 E-9 + 9.996570091213026754 E-61*I
a[-43] = -1.38972254762024517621378 E-9 + 9.388465677448883675 E-61*I
a[-44] = 8.25238360883481060964940 E-10 - 1.3255208419053970184 E-60*I
a[-45] = -3.17325318526524320237867 E-10 - 1.0123287544259977184 E-60*I
a[-46] = -3.76888806832179544950598 E-11 - 1.6683173843410761278 E-60*I
a[-47] = 2.25171487351773085674629 E-10 + 2.2320233303164905300 E-61*I
a[-48] = -2.77091696418487305763715 E-10 + 1.2075997623732215851 E-60*I
a[-49] = 2.42538238284964521710153 E-10 + 1.7521139906568798183 E-60*I
a[-50] = -1.68206588381801979019770 E-10 + 9.312100328125669223 E-61*I
a[-51] = 8.86778193399147529408203 E-11 - 4.800925697999852935 E-61*I
a[-52] = -2.40760544223348638787067 E-11 - 1.1254977818362073943 E-60*I
a[-53] = -1.78663899312889338420433 E-11 - 5.830601395922755530 E-61*I
a[-54] = 3.78952392006332329241532 E-11 + 3.504390548186142454 E-61*I
a[-55] = -4.12278752589250636158352 E-11 + 3.0596921797453011188 E-61*I
a[-56] = 3.43949992116145399144005 E-11 + 8.751393358481279748 E-61*I
a[-57] = -2.32358512986154833240055 E-11 + 2.3083903193421552646 E-61*I
a[-58] = 1.19532569663009671002094 E-11 - 9.758346281421433379 E-61*I
a[-59] = -2.94639468338014118784073 E-12 - 3.420748699541095055 E-61*I
a[-60] = -2.89822830020187414794661 E-12 - 1.0298137458562810148 E-60*I
a[-61] = 5.72765224984859659596865 E-12 + 4.283326808727990317 E-61*I
a[-62] = -6.24100353769722057596258 E-12 + 5.811818892310327139 E-61*I
a[-63] = 5.30699853047606608204884 E-12 - 2.5395251271869018626 E-61*I
a[-64] = -3.71501500370105790077365 E-12 + 9.418938714617855370 E-61*I
a[-65] = 2.05112778246914099072157 E-12 + 9.408049915823806533 E-61*I
a[-66] = -6.68854036779409633895353 E-13 - 1.0120200114991650599 E-60*I
a[-67] = -2.81257305556733932148009 E-13 - 9.543496085046087047 E-61*I
a[-68] = 7.94868549839678126659465 E-13 - 1.0528864539925234357 E-60*I
a[-69] = -9.52953113039993652956814 E-13 - 3.585756009068627262 E-61*I
a[-70] = 8.70430522912296317987430 E-13 + 8.007512533533776157 E-61*I
a[-71] = -6.60146252637862386950784 E-13 + 6.589232225946199111 E-61*I
a[-72] = 4.12498258220042912909922 E-13 + 1.3357319612143506715 E-60*I
a[-73] = -1.87880303840560995974459 E-13 + 0.E-61*I
a[-74] = 1.79009589906311932561769 E-14 + 0.E-61*I
a[-75] = 8.85750482258138756415336 E-14 - 2.1753355395551205912 E-60*I
a[-76] = -1.37721223883570899336952 E-13 - 1.9114560326653858132 E-60*I
a[-77] = 1.43496588961083972036201 E-13 + 1.3386303332392950945 E-60*I
a[-78] = -1.21978846122519073841780 E-13 + 2.1862102205662737612 E-60*I
a[-79] = 8.76469854442097685439913 E-14 + 1.2990126947682685171 E-60*I
a[-80] = -5.14705216926262643814896 E-14 - 1.0018008419977926074 E-60*I
a[-81] = 2.03979604111418701986726 E-14 - 1.3233368591067467349 E-60*I
a[-82] = 2.24192775496443330325499 E-15 - 1.4996906318551638389 E-60*I
a[-83] = -1.58712754588968736068534 E-14 + 8.426085523305792649 E-61*I
a[-84] = 2.16693093743371355272937 E-14 + 1.5376892740692408496 E-60*I
a[-85] = -2.16763197278933496352268 E-14 + 1.4770850422580782269 E-60*I
a[-86] = 1.81095267077642830630561 E-14 + 0.E-61*I
a[-87] = -1.29217532221333541104969 E-14 - 9.882759084708351006 E-61*I
a[-88] = 7.57976874806336744272561 E-15 - 1.5877929675157080387 E-60*I
a[-89] = -3.01138071696685183820669 E-15 - 1.8255317833205893866 E-60*I
a[-90] = -3.37380814768684205008675 E-16 + 1.2143230804412328186 E-60*I
a[-91] = 2.38833050705337079031841 E-15 + 2.570873151632171509 E-60*I
a[-92] = -3.30173234356011298634007 E-15 + 1.2549002395433393497 E-60*I
a[-93] = 3.35871548378515198055450 E-15 + 7.207925042196062712 E-61*I
a[-94] = -2.86994211237359841277488 E-15 + 0.E-61*I
a[-95] = 2.11494906498676096740399 E-15 - 1.4620541430877854840 E-60*I
a[-96] = -1.30987593100428238834775 E-15 - 8.398699513278232340 E-61*I
a[-97] = 5.97641773826568741126775 E-16 + 0.E-61*I
a[-98] = -5.33936990807526121723619 E-17 + 0.E-61*I
a[-99] = -3.01549958759416732988085 E-16 - 1.0449032549761315903 E-60*I
a[-100] = 4.82723205179637784396592 E-16 + 1.1553428412097541002 E-60*I