So few people are working on this problem. I have noticed that results are scattered far and wide. So, I am calculating and compiling data related to the eigenvalues of the Laplacian within polygons. For example, below is a list of the principal Dirichlet eigenvalues within regular polygons (with area Pi, not inscribe in a unit-radius circle), all correctly rounded to 27 decimal places. I actually bounded them to within a relative error of at most 1E-30; and the pentagon, I have to 1E-500. But, below is a good list. The first two are known in closed form, and the last entry is the square of the first root of the Bessel function J_0(x)=0. This last one is the number the sequence is approaching. One can use the formula L = j01^2*(1+4*zeta(3)/N^3+ O(1/N^5)) to estimate the eigenvalue, with improving results as N (the number of polygon sides) increases. I do not use that formula, and I challenge anyone to figure out the non-zero fifth order term. (Incidentally, each eigenvalue shown from 127 sides and up takes about a day of CPU time. I'm doing the 256 sided polygon now, but I can't get thirty digits. I'm at only about 20 digits now,
5.7831876203689428759... where the trailing digits are yet uncertain.) -Bob Jones
<br> 3 7.255197456936871402376313031 <--- 4*Pi/sqrt(3)
<br> 4 6.283185307179586476925286767 <--- 2*Pi
<br> 5 6.022137932042633878298008710
<br> 6 5.917417831613661215688574577
<br> 7 5.866449312655985857712474942
<br> 8 5.838491433592442850516640380
<br> 9 5.821826802270265731735546444
<br> 10 5.811260359219116022788816469
<br> 11 5.804230636717400721878394453
<br> 12 5.799369804356500079315025311
<br> 13 5.795900266856014709790771063
<br> 14 5.793357005271194553273227079
<br> 15 5.791450010651579975693848498
<br> 16 5.789991899990208534349752214
<br> 17 5.788857871981104698617196635
<br> 18 5.787962591857846864212568380
<br> 19 5.787246351381961243008036645
<br> 20 5.786666514140372213530912962
<br> 21 5.786192077596844273028203757
<br> 22 5.785800129428365027574586044
<br> 23 5.785473486454901632048264070
<br> 24 5.785199089790024091834463613
<br> 25 5.784966894130423501418670684
<br> 26 5.784769086314842977992274718
<br> 27 5.784599527236484640593222827
<br> 28 5.784453347751719951196794842
<br> 29 5.784326652365411207380293386
<br> 30 5.784216299392264044119036734
<br> 31 5.784119736080032703344528385
<br> 32 5.784034873702444318330507487
<br> 33 5.783959992040508812335032523
<br> 34 5.783893665694809252033476569
<br> 35 5.783834706770988202840700005
<br> 36 5.783782119955880627699919966
<br> 37 5.783735067049846291962440637
<br> 38 5.783692838773292267706517922
<br> 39 5.783654832210871143386207911
<br> 40 5.783620532655973576951368559
<br> 41 5.783589498912728857243541088
<br> 42 5.783561351331960679963744950
<br> 43 5.783535762021971971277446762
<br> 44 5.783512446799268033474803358
<br> 45 5.783491158538856302913858879
<br> 46 5.783471681656168110105137225
<br> 47 5.783453827508463297379645914
<br> 48 5.783437430546865419250636065
<br> 49 5.783422345083940177286945517
<br> 50 5.783408442568212994780116196
<br> 51 5.783395609277903442166398140
<br> 52 5.783383744362702700019559015
<br> 53 5.783372758175598244048049661
<br> 54 5.783362570847292742897314144
<br> 55 5.783353111064236385644810842
<br> 56 5.783344315018129354638447816
<br> 57 5.783336125500292103090369355
<br> 58 5.783328491118809153913672275
<br> 59 5.783321365620033853737939611
<br> 60 5.783314707299059428210797800
<br> 61 5.783308478486244250571058736
<br> 62 5.783302645098928410170380598
<br> 63 5.783297176249175699422050587
<br> 64 5.783292043899785027461125829
<br> 65 5.783287222561990216101379319
<br> 66 5.783282689029249211147849046
<br> 67 5.783278422142346980297970170
<br> 68 5.783274402581728387648667337
<br> 69 5.783270612683560625466506793
<br> 70 5.783267036276517720104953505
<br> 71 5.783263658536697267505357057
<br> 72 5.783260465858434270390794564
<br> 73 5.783257445739078946040870361
<br> 74 5.783254586676063084321584559
<br> 75 5.783251878074799951907284413
<br> 76 5.783249310166151671634629193
<br> 77 5.783246873932360298530618523
<br> 78 5.783244561040478512305563119
<br> 79 5.783242363782456338656120879
<br> 80 5.783240275021144443460263290
<br> 81 5.783238288141564708788818914
<br> 82 5.783236397006877016631166830
<br> 83 5.783234595918539141935169159
<br> 84 5.783232879580215836619574007
<br> 85 5.783231243065044797761869160
<br> 86 5.783229681785912299987294190
<br> 87 5.783228191468430723800997531
<br> 127 5.783199538123680412174552014
<br> 128 5.783199222432098956985238320
<br> 129 5.783198916453726829015452454
<br> 130 5.783198619817847494322697718
<br> 5.783185962946784521175995758 <-- j_{0,1}^2 (J_0(j_{0,1})=0)