By: Robert S. Jones
rsjones7 at yahoo dot com
June, 2015
The lowest Dirichlet eigenvalues of the Laplacian for the regular polygons, with number of sides $S$ and area $\pi$, are presented in the following table. These are eigenvalue solutions of $$\nabla^2\Phi(k;\mathbf{r})+k^2\Phi(k;\mathbf{r})=0$$ subject to the boundary condition $$\Phi(k;\mathbf{r})=0\mbox{ for } \mathbf{r}\in\partial\Omega$$ where $\partial\Omega$ is the boundary of the regular polygon $\Omega$. The eigenvalue is $\lambda=k^2$.
The calculation becomes more and more difficult for increasing values of $S$. At $S=130$, the calculation takes about one day of CPU time (i7, quad-core, 8 threads, 8GB RAM). For low values of $S$, the convergence rate is quite good allowing me, for example, to calculate the the regular pentagon lowest Dirichlet eigenvalue to 1000 digits (which took about 2.7 days CPU time, after increasing the RAM to 16 GB).
Upper and lower bounds are presented such that all of the eigenvalues have a relative error of $\epsilon<10^{-30}$. The notation is such that if $\lambda=1.23_{34}^{43}$ then $1.2334<\lambda<1.2343$.
The last two columns list the approximate number of correct digits in the estimate $\tilde\lambda_a$ using the asymptotic formulas (see below, here $a=4,5,6$ is the highest order of the expansion). That number is $\widetilde{D}_a=-\log_{10}(|\tilde\lambda_a-\lambda|/\lambda)$ where $\lambda$ is the exact value.
The first two, the equilateral triangle and the square, of course, have closed-form results, $\lambda(3)=4\,\pi/\sqrt{3}$ and $\lambda(4)=2\,\pi$
The last row is the unit-radius circle eigenvalue, $j_{0,1}^2$.
Two text files with these 30-digit results for the pentagon to the 130-sided, $\pi$-area regular polygon are provided here:
- Lower Bounds (beginning with 6.022...1005402)
- Upper Bounds (beginning with 6.022...1005592)
$S$ $\lambda$ $\epsilon$ $\widetilde{D}_4$ $\widetilde{D}_5$ $\widetilde{D}_6$ 3 $7.25519745693687140237631303056862$ $ 1.215$ $ 1.226$ $ 1.988$ 4 $6.28318530717958647692528676655901$ $ 1.982$ $ 1.999$ $ 4.068$ 5 $6.02213793204263387829800871005_{402}^{592}$ $3.13\times 10^{-31}$ $ 2.562$ $ 2.585$ $ 3.786$ 6 $5.9174178316136612156885745768_{3851}^{4127}$ $4.66\times 10^{-31}$ $ 3.032$ $ 3.059$ $ 4.155$ 7 $5.8664493126559858577124749417_{5839}^{6058}$ $3.71\times 10^{-31}$ $ 3.427$ $ 3.459$ $ 4.532$ 8 $5.83849143359244285051664037956_{027}^{492}$ $7.94\times 10^{-31}$ $ 3.768$ $ 3.804$ $ 4.882$ 9 $5.82182680227026573173554644371_{507}^{762}$ $4.36\times 10^{-31}$ $ 4.069$ $ 4.109$ $ 5.202$ 10 $5.81126035921911602278881646881_{013}^{374}$ $6.20\times 10^{-31}$ $ 4.337$ $ 4.381$ $ 5.494$ 11 $5.80423063671740072187839445285_{306}^{754}$ $7.70\times 10^{-31}$ $ 4.579$ $ 4.628$ $ 5.762$ 12 $5.7993698043565000793150253110_{0613}^{1122}$ $8.77\times 10^{-31}$ $ 4.800$ $ 4.852$ $ 6.009$ 13 $5.79590026685601470979077106333_{349}^{895}$ $9.41\times 10^{-31}$ $ 5.003$ $ 5.059$ $ 6.239$ 14 $5.79335700527119455327322707868_{188}^{752}$ $9.71\times 10^{-31}$ $ 5.191$ $ 5.251$ $ 6.452$ 15 $5.7914500106515799756938484981_{4604}^{5169}$ $9.74\times 10^{-31}$ $ 5.365$ $ 5.429$ $ 6.652$ 16 $5.7899918999902085343497522138_{2606}^{3161}$ $9.57\times 10^{-31}$ $ 5.528$ $ 5.596$ $ 6.840$ 17 $5.7888578719811046986171966351_{8657}^{9195}$ $9.27\times 10^{-31}$ $ 5.682$ $ 5.753$ $ 7.017$ 18 $5.78796259185784686421256838015_{194}^{509}$ $5.42\times 10^{-31}$ $ 5.826$ $ 5.901$ $ 7.184$ 19 $5.78724635138196124300803664483_{173}^{663}$ $8.45\times 10^{-31}$ $ 5.962$ $ 6.041$ $ 7.343$ 20 $5.78666651414037221353091296202_{392}^{856}$ $8.00\times 10^{-31}$ $ 6.091$ $ 6.174$ $ 7.494$ 21 $5.7861920775968442730282037573_{2687}^{3124}$ $7.54\times 10^{-31}$ $ 6.214$ $ 6.300$ $ 7.637$ 22 $5.78580012942836502757458604434_{176}^{587}$ $7.08\times 10^{-31}$ $ 6.331$ $ 6.420$ $ 7.775$ 23 $5.785473486454901632048264070_{19447}^{20012}$ $9.76\times 10^{-31}$ $ 6.442$ $ 6.536$ $ 7.906$ 24 $5.7851990897900240918344636129_{7661}^{8182}$ $8.99\times 10^{-31}$ $ 6.549$ $ 6.646$ $ 8.032$ 25 $5.78496689413042350141867068388_{314}^{794}$ $8.29\times 10^{-31}$ $ 6.651$ $ 6.752$ $ 8.153$ 26 $5.78476908631484297799227471769_{006}^{450}$ $7.65\times 10^{-31}$ $ 6.749$ $ 6.853$ $ 8.270$ 27 $5.78459952723648464059322282715_{248}^{815}$ $9.79\times 10^{-31}$ $ 6.844$ $ 6.951$ $ 8.382$ 28 $5.784453347751719951196794842_{39684}^{40203}$ $8.95\times 10^{-31}$ $ 6.935$ $ 7.045$ $ 8.490$ 29 $5.78432665236541120738029338594_{357}^{832}$ $8.19\times 10^{-31}$ $ 7.022$ $ 7.136$ $ 8.594$ 30 $5.78421629939226404411903673407_{022}^{458}$ $7.52\times 10^{-31}$ $ 7.107$ $ 7.224$ $ 8.695$ 31 $5.784119736080032703344528384_{59653}^{60185}$ $9.18\times 10^{-31}$ $ 7.189$ $ 7.309$ $ 8.793$ 32 $5.7840348737024443183305074869_{2753}^{3239}$ $8.38\times 10^{-31}$ $ 7.268$ $ 7.392$ $ 8.888$ 33 $5.7839599920405088123350325226_{2669}^{3114}$ $7.66\times 10^{-31}$ $ 7.344$ $ 7.471$ $ 8.979$ 34 $5.78389366569480925203347656871_{102}^{629}$ $9.10\times 10^{-31}$ $ 7.418$ $ 7.549$ $ 9.069$ 35 $5.78383470677098820284070000522_{066}^{547}$ $8.30\times 10^{-31}$ $ 7.490$ $ 7.624$ $ 9.155$ 36 $5.7837821199558806276999199658_{4812}^{5374}$ $9.70\times 10^{-31}$ $ 7.560$ $ 7.697$ $ 9.239$ 37 $5.78373506704984629196244063652_{200}^{712}$ $8.83\times 10^{-31}$ $ 7.628$ $ 7.768$ $ 9.321$ 38 $5.7836928387732922677065179222_{5965}^{6432}$ $8.06\times 10^{-31}$ $ 7.694$ $ 7.837$ $ 9.401$ 39 $5.78365483221087114338620791091_{180}^{716}$ $9.24\times 10^{-31}$ $ 7.759$ $ 7.905$ $ 9.479$ 40 $5.7836205326559735769513685588_{8691}^{9180}$ $8.43\times 10^{-31}$ $ 7.821$ $ 7.970$ $ 9.555$ 41 $5.7835894989127288572435410875_{3914}^{4468}$ $9.56\times 10^{-31}$ $ 7.882$ $ 8.035$ $ 9.629$ 42 $5.7835613513319606799637449504_{6674}^{7179}$ $8.72\times 10^{-31}$ $ 7.942$ $ 8.097$ $ 9.701$ 43 $5.78353576202197197127744676206_{295}^{862}$ $9.78\times 10^{-31}$ $ 8.000$ $ 8.158$ $ 9.771$ 44 $5.7835124467992680334748033579_{8534}^{9051}$ $8.92\times 10^{-31}$ $ 8.056$ $ 8.218$ $ 9.840$ 45 $5.78349115853885630291385887880_{169}^{744}$ $9.91\times 10^{-31}$ $ 8.112$ $ 8.276$ $ 9.908$ 46 $5.78347168165616811010513722523_{471}^{995}$ $9.05\times 10^{-31}$ $ 8.166$ $ 8.333$ $ 9.974$ 47 $5.78345382750846329737964591391_{342}^{920}$ $9.98\times 10^{-31}$ $ 8.219$ $ 8.389$ $10.038$ 48 $5.78343743054686541925063606475_{348}^{877}$ $9.12\times 10^{-31}$ $ 8.271$ $ 8.444$ $10.101$ 49 $5.78342234508394017728694551658_{365}^{944}$ $9.99\times 10^{-31}$ $ 8.321$ $ 8.497$ $10.163$ 50 $5.78340844256821299478011619632_{077}^{606}$ $9.14\times 10^{-31}$ $ 8.371$ $ 8.550$ $10.224$ 51 $5.78339560927790344216639813994_{232}^{718}$ $8.38\times 10^{-31}$ $ 8.419$ $ 8.601$ $10.284$ 52 $5.7833837443627027000195590153_{1956}^{2483}$ $9.11\times 10^{-31}$ $ 8.467$ $ 8.652$ $10.342$ 53 $5.7833727581755982440480496612_{8572}^{9056}$ $8.36\times 10^{-31}$ $ 8.514$ $ 8.701$ $10.399$ 54 $5.78336257084729274289731414400_{023}^{546}$ $9.04\times 10^{-31}$ $ 8.559$ $ 8.750$ $10.455$ 55 $5.78335311106423638564481084216_{098}^{579}$ $8.31\times 10^{-31}$ $ 8.604$ $ 8.797$ $10.511$ 56 $5.7833443150181293546384478158_{8517}^{9035}$ $8.94\times 10^{-31}$ $ 8.648$ $ 8.844$ $10.565$ 57 $5.78333612550029210309036935466_{229}^{706}$ $8.22\times 10^{-31}$ $ 8.691$ $ 8.890$ $10.618$ 58 $5.7833284911188091539136722748_{1900}^{2410}$ $8.82\times 10^{-31}$ $ 8.734$ $ 8.935$ $10.670$ 59 $5.78332136562003385373793961146_{241}^{711}$ $8.12\times 10^{-31}$ $ 8.776$ $ 8.980$ $10.722$ 60 $5.78331470729905942821079780018_{166}^{669}$ $8.67\times 10^{-31}$ $ 8.817$ $ 9.024$ $10.772$ 61 $5.7833084784862442505710587362_{1718}^{2253}$ $9.23\times 10^{-31}$ $ 8.857$ $ 9.066$ $10.822$ 62 $5.78330264509892841017038059784_{138}^{705}$ $9.80\times 10^{-31}$ $ 8.896$ $ 9.109$ $10.871$ 63 $5.7832971762491756994220505872_{0528}^{1051}$ $9.02\times 10^{-31}$ $ 8.935$ $ 9.150$ $10.919$ 64 $5.7832920438997850274611258286_{1636}^{2189}$ $9.55\times 10^{-31}$ $ 8.974$ $ 9.191$ $10.967$ 65 $5.7832872225619902161013793189_{5601}^{6111}$ $8.81\times 10^{-31}$ $ 9.011$ $ 9.232$ $11.014$ 66 $5.7832826890292492111478490456_{5885}^{6424}$ $9.30\times 10^{-31}$ $ 9.048$ $ 9.271$ $11.060$ 67 $5.7832784221423469802979701702_{2974}^{3472}$ $8.59\times 10^{-31}$ $ 9.085$ $ 9.310$ $11.105$ 68 $5.7832744025817283876486673365_{4675}^{5199}$ $9.04\times 10^{-31}$ $ 9.121$ $ 9.349$ $11.150$ 69 $5.78327061268356062546650679310_{308}^{858}$ $9.50\times 10^{-31}$ $ 9.156$ $ 9.387$ $11.194$ 70 $5.7832670362765177201049535052_{8863}^{9439}$ $9.95\times 10^{-31}$ $ 9.191$ $ 9.424$ $11.237$ 71 $5.7832636585366972675053570571_{5508}^{6041}$ $9.20\times 10^{-31}$ $ 9.225$ $ 9.461$ $11.280$ 72 $5.78326046585843427039079456440_{041}^{599}$ $9.63\times 10^{-31}$ $ 9.259$ $ 9.497$ $11.322$ 73 $5.78325744573907894604087036089_{114}^{630}$ $8.92\times 10^{-31}$ $ 9.293$ $ 9.533$ $11.363$ 74 $5.7832545866760630843215845592_{6763}^{7303}$ $9.31\times 10^{-31}$ $ 9.325$ $ 9.569$ $11.404$ 75 $5.78325187807479995190728441292_{112}^{674}$ $9.70\times 10^{-31}$ $ 9.358$ $ 9.604$ $11.445$ 76 $5.7832493101661516716346291929_{1870}^{2391}$ $9.00\times 10^{-31}$ $ 9.390$ $ 9.638$ $11.485$ 77 $5.78324687393236029853061852313_{162}^{704}$ $9.36\times 10^{-31}$ $ 9.421$ $ 9.672$ $11.524$ 78 $5.7832445610404785123055631187_{1525}^{2088}$ $9.72\times 10^{-31}$ $ 9.453$ $ 9.705$ $11.563$ 79 $5.7832423637824563386561208792_{0871}^{1395}$ $9.03\times 10^{-31}$ $ 9.483$ $ 9.739$ $11.602$ 80 $5.78324027502114444346026329033_{075}^{617}$ $9.36\times 10^{-31}$ $ 9.514$ $ 9.771$ $11.640$ 81 $5.78323828814156470878881891374_{106}^{668}$ $9.70\times 10^{-31}$ $ 9.543$ $ 9.804$ $11.677$ 82 $5.7832363970068770166311668301_{4963}^{5486}$ $9.02\times 10^{-31}$ $ 9.573$ $ 9.836$ $11.714$ 83 $5.78323459591853914193516915928_{038}^{579}$ $9.33\times 10^{-31}$ $ 9.602$ $ 9.867$ $11.751$ 84 $5.78323287958021583661957400684_{188}^{746}$ $9.64\times 10^{-31}$ $ 9.631$ $ 9.898$ $11.787$ 85 $5.7832312430650447977618691596_{8600}^{9176}$ $9.95\times 10^{-31}$ $ 9.659$ $ 9.929$ $11.823$ 86 $5.78322968178591229998729418953_{020}^{557}$ $9.26\times 10^{-31}$ $ 9.687$ $ 9.959$ $11.858$ 87 $5.78322819146843072380099753098_{290}^{843}$ $9.55\times 10^{-31}$ $ 9.715$ $ 9.989$ $11.893$ 88 $5.7832267681263447879041212711_{1715}^{2285}$ $9.83\times 10^{-31}$ $ 9.743$ $10.019$ $11.927$ 89 $5.78322540803912364438644198743_{300}^{831}$ $9.17\times 10^{-31}$ $ 9.770$ $10.049$ $11.962$ 90 $5.78322410773152267822276798164_{137}^{683}$ $9.43\times 10^{-31}$ $ 9.796$ $10.078$ $11.995$ 91 $5.783222863954922345132559440_{09855}^{10417}$ $9.69\times 10^{-31}$ $ 9.823$ $10.106$ $12.029$ 92 $5.7832216736702720961404775421_{0778}^{1354}$ $9.95\times 10^{-31}$ $ 9.849$ $10.135$ $12.062$ 93 $5.7832205340324857277389860275_{8692}^{9230}$ $9.29\times 10^{-31}$ $ 9.875$ $10.163$ $12.094$ 94 $5.783219442376150669518408021_{29845}^{30398}$ $9.53\times 10^{-31}$ $ 9.901$ $10.191$ $12.127$ 95 $5.78321839620242804116261639329_{040}^{606}$ $9.77\times 10^{-31}$ $ 9.926$ $10.218$ $12.159$ 96 $5.783217393167033007010852241_{39585}^{40114}$ $9.13\times 10^{-31}$ $ 9.951$ $10.245$ $12.190$ 97 $5.7832164310691962277585187415_{2773}^{3315}$ $9.36\times 10^{-31}$ $ 9.976$ $10.272$ $12.222$ 98 $5.78321550784151722801068379624_{253}^{808}$ $9.57\times 10^{-31}$ $10.000$ $10.299$ $12.253$ 99 $5.7832146215406294155617881961_{5703}^{6270}$ $9.79\times 10^{-31}$ $10.024$ $10.326$ $12.283$ 100 $5.7832137703386044343671448760_{6733}^{7265}$ $9.17\times 10^{-31}$ $10.048$ $10.352$ $12.314$ 101 $5.7832129525150306224265566440_{5557}^{6099}$ $9.37\times 10^{-31}$ $10.072$ $10.378$ $12.344$ 102 $5.7832121664497066780323274791_{2494}^{3048}$ $9.57\times 10^{-31}$ $10.096$ $10.403$ $12.374$ 103 $5.78321141061589730038223792298_{352}^{917}$ $9.76\times 10^{-31}$ $10.119$ $10.429$ $12.403$ 104 $5.7832106835741026399496287805_{6681}^{7258}$ $9.96\times 10^{-31}$ $10.142$ $10.454$ $12.432$ 105 $5.7832099839662979373801765921_{2700}^{3241}$ $9.34\times 10^{-31}$ $10.165$ $10.479$ $12.461$ 106 $5.7832093105106038060342151456_{8977}^{9528}$ $9.52\times 10^{-31}$ $10.187$ $10.503$ $12.490$ 107 $5.7832086619963512745011282308_{8693}^{9255}$ $9.70\times 10^{-31}$ $10.209$ $10.528$ $12.518$ 108 $5.78320803727950899717733132378_{083}^{655}$ $9.87\times 10^{-31}$ $10.232$ $10.552$ $12.546$ 109 $5.7832074352784430036203824301_{5931}^{6468}$ $9.27\times 10^{-31}$ $10.254$ $10.576$ $12.574$ 110 $5.78320685496998202644834606904_{166}^{713}$ $9.43\times 10^{-31}$ $10.275$ $10.600$ $12.602$ 111 $5.7832062953857638544958683846_{8888}^{9402}$ $8.87\times 10^{-31}$ $10.297$ $10.623$ $12.629$ 112 $5.783205755608840330598899278_{69493}^{70058}$ $9.75\times 10^{-31}$ $10.318$ $10.646$ $12.656$ 113 $5.7832052347705205764174624901_{3556}^{4130}$ $9.91\times 10^{-31}$ $10.339$ $10.670$ $12.683$ 114 $5.7832047320474338019919255227_{7639}^{8179}$ $9.32\times 10^{-31}$ $10.360$ $10.693$ $12.710$ 115 $5.7832042466587946646835703161_{8765}^{9314}$ $9.46\times 10^{-31}$ $10.381$ $10.715$ $12.736$ 116 $5.78320377786385559803797727272_{276}^{833}$ $9.61\times 10^{-31}$ $10.401$ $10.738$ $12.763$ 117 $5.7832033249595318512911938361_{7560}^{8084}$ $9.05\times 10^{-31}$ $10.422$ $10.760$ $12.788$ 118 $5.7832028872781861783972732175_{4920}^{5452}$ $9.18\times 10^{-31}$ $10.442$ $10.782$ $12.814$ 119 $5.7832024641855612037927177369_{4957}^{5497}$ $9.31\times 10^{-31}$ $10.462$ $10.804$ $12.840$ 120 $5.7832020550788484815190743450_{7680}^{8227}$ $9.44\times 10^{-31}$ $10.482$ $10.826$ $12.865$ 121 $5.78320165938488416452948767025_{078}^{632}$ $9.57\times 10^{-31}$ $10.501$ $10.848$ $12.890$ 122 $5.78320127655846202071020955606_{248}^{810}$ $9.69\times 10^{-31}$ $10.521$ $10.869$ $12.915$ 123 $5.78320090608075527914348181390_{019}^{588}$ $9.81\times 10^{-31}$ $10.540$ $10.890$ $12.940$ 124 $5.7832005474578394714033936689_{3655}^{4230}$ $9.93\times 10^{-31}$ $10.559$ $10.911$ $12.964$ 125 $5.78320020021930905447076146118_{363}^{906}$ $9.37\times 10^{-31}$ $10.578$ $10.932$ $12.989$ 126 $5.7831998639169811697955997275_{4738}^{5287}$ $9.48\times 10^{-31}$ $10.597$ $10.953$ $13.013$ 127 $5.7831995381236804121745520138_{6591}^{7147}$ $9.59\times 10^{-31}$ $10.615$ $10.973$ $13.037$ 128 $5.78319922243209895698523832013_{030}^{555}$ $9.06\times 10^{-31}$ $10.634$ $10.994$ $13.060$ 129 $5.78319891645372682901545245421_{112}^{643}$ $9.16\times 10^{-31}$ $10.652$ $11.014$ $13.084$ 130 $5.78319861981784749432269771828_{013}^{587}$ $9.91\times 10^{-31}$ $10.671$ $11.034$ $13.107$ $\infty$ $5.783185962946784521175995758455...$ --- --- --- ---
Estimates of these eigenvalues can be obtained using the asymptotic formula: $$\tilde\lambda_4=j_{0,1}^2\left[1+\frac{4\,\zeta(3)}{S^3} +\mathcal{O}\left(\frac{1}{S^5}\right)\right]$$ $$\tilde\lambda_5=j_{0,1}^2\left[1+\frac{4\,\zeta(3)}{S^3} +\frac{\left(12-2j_{0,1}^2\right)\zeta(5)}{S^5} +\mathcal{O}\left(\frac{1}{S^6}\right)\right]$$ $$\tilde\lambda_6=j_{0,1}^2\left[1+\frac{4\,\zeta(3)}{S^3} +\frac{\left(12-2\,j_{0,1}^2\right)\zeta(5)}{S^5} +\frac{\left(8+4\,j_{0,1}^2\right)\zeta^2(3)}{S^6} +\mathcal{O}\left(\frac{1}{S^7}\right) \right] $$ where $j_{0,1}$ is the first root of the Bessel function $J_0(x)$, and $\zeta(x)$ is the Riemann-Zeta function. This formula, which was not used to calculate my results (second column), does yield increasingly better results, achieving 30-digit precision for $S>20,000$. ($S>10^{30/7}$)
Thank you Mark Boady for offering asymptotic terms beyond the fourth order.
As an aside, the over 30-digit result for the lowest Dirichlet eigenvalue of the famous but somewhat obscure 65537-sided regular polygon (widipedia 65537-gon), with area $\pi$, is very close to $$ \frac{\lambda_1}{j_{01}^2} = 1.000000000000017081474070818627484... $$ $$ \lambda_1 = 5.783185962946883306517068556211370... $$ where the above formula for $\tilde\lambda_6$ above simply used, and there is some uncertainty in the final one or two digits.