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Neumann eigenvalues of the classical planar domains
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Numbers
$\Omega$
$n$ 
$\mu_n(\Omega)$
unit square
1:
0
unit square
2:
9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
unit square
3:
9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
equals: #unit-square,2
unit square
4:
19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#unit-square,1
unit square
5:
39.47841760435743447533796399950460454125479762896316250565339750488017928967682097200709361487420893
unit square
6:
39.47841760435743447533796399950460454125479762896316250565339750488017928967682097200709361487420893
equals: #unit-square,5
unit square
7:
49.34802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116
unit square
8:
49.34802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116
equals: #unit-square,7
unit square
9:
78.95683520871486895067592799900920908250959525792632501130679500976035857935364194401418722974841785
unit square
10:
88.82643960980422756951041899888536021782329466516711563772014438598040340177284718701596063346697009
unit square
11:
88.82643960980422756951041899888536021782329466516711563772014438598040340177284718701596063346697009
equals: #unit-square,10
unit square
12:
98.69604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
unit square
13:
98.69604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
equals: #unit-square,12
unit square
14:
128.3048572141616620448483829983899647590780922941302781433735418908605826914496681590230542483411790
unit square
15:
128.3048572141616620448483829983899647590780922941302781433735418908605826914496681590230542483411790
equals: #unit-square,14
unit square
16:
157.9136704174297379013518559980184181650191905158526500226135900195207171587072838880283744594968357
unit square
17:
157.9136704174297379013518559980184181650191905158526500226135900195207171587072838880283744594968357
equals: #unit-square,16
unit square
18:
167.7832748185190965201863469978945693003328899230934406490269393957407619811264891310301478632153879
unit square
19:
167.7832748185190965201863469978945693003328899230934406490269393957407619811264891310301478632153879
equals: #unit-square,18
unit square
20:
177.6528792196084551390208379977707204356465893303342312754402887719608068035456943740319212669339402
$1\times2$ rectangle
1:
0
equals: #unit-square,1
$1\times2$ rectangle
2:
2.467401100272339654708622749969037783828424851810197656603337344055011205604801310750443350929638058
$1\times2$ rectangle
3:
9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
equals: #unit-square,2
$1\times2$ rectangle
4:
9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
equals: #unit-square,2
$1\times2$ rectangle
5:
12.33700550136169827354311374984518891914212425905098828301668672027505602802400655375221675464819029
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#rectangle-1x2,1
$1\times2$ rectangle
6:
19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
equals: #unit-square,4
$1\times2$ rectangle
7:
22.20660990245105689237760474972134005445582366629177890943003609649510085044321179675399015836674252
$1\times2$ rectangle
8:
32.07621430354041551121209574959749118976952307353256953584338547271514567286241703975576356208529475
$1\times2$ rectangle
9:
39.47841760435743447533796399950460454125479762896316250565339750488017928967682097200709361487420893
equals: #unit-square,5
$1\times2$ rectangle
10:
39.47841760435743447533796399950460454125479762896316250565339750488017928967682097200709361487420893
equals: #unit-square,5
$1\times2$ rectangle
11:
41.94581870462977413004658674947364232508322248077336016225673484893519049528162228275753696580384699
$1\times2$ rectangle
12:
49.34802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116
equals: #unit-square,7
$1\times2$ rectangle
13:
49.34802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116
equals: #unit-square,7
$1\times2$ rectangle
14:
61.68502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145
$1\times2$ rectangle
15:
61.68502750680849136771556874922594459571062129525494141508343360137528014012003276876108377324095145
equals: #rectangle-1x2,14
$1\times2$ rectangle
16:
71.55463190789784998655005974910209573102432070249573204149678297759532496253923801176285717695950368
$1\times2$ rectangle
17:
78.95683520871486895067592799900920908250959525792632501130679500976035857935364194401418722974841785
equals: #unit-square,9
$1\times2$ rectangle
18:
88.82643960980422756951041899888536021782329466516711563772014438598040340177284718701596063346697009
equals: #unit-square,10
$1\times2$ rectangle
19:
88.82643960980422756951041899888536021782329466516711563772014438598040340177284718701596063346697009
equals: #unit-square,10
$1\times2$ rectangle
20:
91.29384071007656722421904174885439800165171951697731329432348173003541460737764849776640398439660814
unit disk
1:
0
equals: #unit-square,1
unit disk
2:
3.389957716671888726864490927467778458229991053314719642831603773491653801489262289113456408961533374
unit disk
3:
3.389957716671888726864490927467778458229991053314719642831603773491653801489262289113456408961533374
equals: #unit-disk,2
unit disk
4:
9.328363213746357907249605773914062257025643293087900766802056378151667962215237953758078543411296761
unit disk
5:
9.328363213746357907249605773914062257025643293087900766802056378151667962215237953758078543411296761
equals: #unit-disk,4
unit disk
6:
14.68197064212389325721977776863010698968158698215497813417558917182975765058406195920176886364035598
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#half-disk,1
unit disk
7:
17.64998851974964146094387891607392914293923917998284104344787846675353915623646392715745445000518405
unit disk
8:
17.64998851974964146094387891607392914293923917998284104344787846675353915623646392715745445000518405
equals: #unit-disk,7
unit disk
9:
28.27637124872566071668905621948688666607820182013756094645099211376215522977615154416486450827229152
unit disk
10:
28.27637124872566071668905621948688666607820182013756094645099211376215522977615154416486450827229152
equals: #unit-disk,9
unit disk
11:
28.42428204737229244428450191056092135755229385859919793905546164013334807376271920600834388594385225
unit disk
12:
28.42428204737229244428450191056092135755229385859919793905546164013334807376271920600834388594385225
equals: #unit-disk,11
unit disk
13:
41.16013348015308667553383008120460476344550230693859677695135901207226963622482849719890130770123081
unit disk
14:
41.16013348015308667553383008120460476344550230693859677695135901207226963622482849719890130770123081
equals: #unit-disk,13
unit disk
15:
44.97222241779393792230076068346165021320638715002280422246216415545071155748786745423953783773781595
unit disk
16:
44.97222241779393792230076068346165021320638715002280422246216415545071155748786745423953783773781595
equals: #unit-disk,15
unit disk
17:
49.21845632169460367026708284638813893242541182900521313164150592823154609982169550170684128352660434
unit disk
18:
56.26899377338456659780758834916747443968632056078928852391575979684732776741405839627843671578262501
unit disk
19:
56.26899377338456659780758834916747443968632056078928852391575979684732776741405839627843671578262501
equals: #unit-disk,18
unit disk
20:
64.24401772794530547757332111633357732773554017206385868899311271928724821568582707635926704007557318
half disk
1:
0
equals: #unit-square,1
half disk
2:
3.389957716671888726864490927467778458229991053314719642831603773491653801489262289113456408961533374
equals: #unit-disk,2
half disk
3:
9.328363213746357907249605773914062257025643293087900766802056378151667962215237953758078543411296761
equals: #unit-disk,4
half disk
4:
14.68197064212389325721977776863010698968158698215497813417558917182975765058406195920176886364035598
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#half-disk,1
half disk
5:
17.64998851974964146094387891607392914293923917998284104344787846675353915623646392715745445000518405
equals: #unit-disk,7
half disk
6:
28.27637124872566071668905621948688666607820182013756094645099211376215522977615154416486450827229152
equals: #unit-disk,9
half disk
7:
28.42428204737229244428450191056092135755229385859919793905546164013334807376271920600834388594385225
equals: #unit-disk,11
half disk
8:
41.16013348015308667553383008120460476344550230693859677695135901207226963622482849719890130770123081
equals: #unit-disk,13
half disk
9:
44.97222241779393792230076068346165021320638715002280422246216415545071155748786745423953783773781595
equals: #unit-disk,15
half disk
10:
49.21845632169460367026708284638813893242541182900521313164150592823154609982169550170684128352660434
equals: #unit-disk,17
half disk
11:
56.26899377338456659780758834916747443968632056078928852391575979684732776741405839627843671578262501
equals: #unit-disk,18
half disk
12:
64.24401772794530547757332111633357732773554017206385868899311271928724821568582707635926704007557318
equals: #unit-disk,20
half disk
13:
72.86869710635145682642463974594380210164952956352558474286077361007326061760907715372437580703145851
half disk
14:
73.57927884427026929646936816167783287235391020416460773268256559758593879290940460215239829574324853
half disk
15:
86.16288079626728429694875952089635395381122480137407784224787333604029164771416004505480617708637538
half disk
16:
93.07274453142470833175125085959450039476717495767639922019976435577922603586368665044927365598340649
half disk
17:
99.39028867557892621206692655961362610004700285187705786838917290648202285321993235633587204227937028
half disk
18:
103.4994538951365803322236325356130557498350227148762554092356979341276008828238257665153101012003074
half disk
19:
110.6674728035254686092669998033403656776873492451796453786284515912855712174131794134456200912864032
half disk
20:
114.7348177086648001620969594292200650485437821401245495770095141005111666828949371544859894652695957
quarter disk
1:
0
equals: #unit-square,1
quarter disk
2:
9.328363213746357907249605773914062257025643293087900766802056378151667962215237953758078543411296761
equals: #unit-disk,4
quarter disk
3:
14.68197064212389325721977776863010698968158698215497813417558917182975765058406195920176886364035598
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#half-disk,1
quarter disk
4:
28.27637124872566071668905621948688666607820182013756094645099211376215522977615154416486450827229152
equals: #unit-disk,9
quarter disk
5:
44.97222241779393792230076068346165021320638715002280422246216415545071155748786745423953783773781595
equals: #unit-disk,15
quarter disk
6:
49.21845632169460367026708284638813893242541182900521313164150592823154609982169550170684128352660434
equals: #unit-disk,17
quarter disk
7:
56.26899377338456659780758834916747443968632056078928852391575979684732776741405839627843671578262501
equals: #unit-disk,18
quarter disk
8:
86.16288079626728429694875952089635395381122480137407784224787333604029164771416004505480617708637538
equals: #half-disk,15
quarter disk
9:
93.07274453142470833175125085959450039476717495767639922019976435577922603586368665044927365598340649
equals: #half-disk,16
quarter disk
10:
99.39028867557892621206692655961362610004700285187705786838917290648202285321993235633587204227937028
equals: #half-disk,17
quarter disk
11:
103.4994538951365803322236325356130557498350227148762554092356979341276008828238257665153101012003074
equals: #half-disk,18
quarter disk
12:
137.7087218220143610390272213830428255500476670063621201006042212580973280840784018072562060793063325
quarter disk
13:
138.5535376970133764229891579612753947325230746414815286950363934831605737119337247716615678298603005
quarter disk
14:
160.8308017474755842781313887248219958892038518846982742621654319178208663736105445822731995467869968
quarter disk
15:
173.4586684849988664554502752573828506432659037059922580171177265187983218814527664816102458145327020
quarter disk
16:
177.5207668138046498617171280731799337892088318616092849225991982121459633924831517733000012458202548
quarter disk
17:
192.6222849532841138043680629240578789462007259282122787793906814770757115261366109840734734643044826
quarter disk
18:
199.2478740391304803780248367737372650502633948671017333063499427919367622545710607830923058428133542
quarter disk
19:
233.1173651290127939828581901699426670984103262857560426931111914793498254547127756078494115732905926
quarter disk
20:
254.8527135121500302010689155037659767110663274994004172185347656793699565638394525175810092783794481
circular sector of angle $\pi/3$
1:
0
equals: #unit-square,1
circular sector of angle $\pi/3$
2:
14.68197064212389325721977776863010698968158698215497813417558917182975765058406195920176886364035598
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#half-disk,1
circular sector of angle $\pi/3$
3:
17.64998851974964146094387891607392914293923917998284104344787846675353915623646392715745445000518405
equals: #unit-disk,7
circular sector of angle $\pi/3$
4:
49.21845632169460367026708284638813893242541182900521313164150592823154609982169550170684128352660434
equals: #unit-disk,17
circular sector of angle $\pi/3$
5:
56.26899377338456659780758834916747443968632056078928852391575979684732776741405839627843671578262501
equals: #unit-disk,18
circular sector of angle $\pi/3$
6:
64.24401772794530547757332111633357732773554017206385868899311271928724821568582707635926704007557318
equals: #unit-disk,20
circular sector of angle $\pi/3$
7:
103.4994538951365803322236325356130557498350227148762554092356979341276008828238257665153101012003074
equals: #half-disk,18
circular sector of angle $\pi/3$
8:
114.7348177086648001620969594292200650485437821401245495770095141005111666828949371544859894652695957
equals: #half-disk,20
circular sector of angle $\pi/3$
9:
128.7299984651091667049953324205398356660515284929530848100000656687094187346954972210263271705632218
circular sector of angle $\pi/3$
10:
137.7087218220143610390272213830428255500476670063621201006042212580973280840784018072562060793063325
equals: #quarter-disk,12
circular sector of angle $\pi/3$
11:
177.5207668138046498617171280731799337892088318616092849225991982121459633924831517733000012458202548
equals: #quarter-disk,16
circular sector of angle $\pi/3$
12:
192.6222849532841138043680629240578789462007259282122787793906814770757115261366109840734734643044826
equals: #quarter-disk,17
circular sector of angle $\pi/3$
13:
212.7469702270816336543022578905874682651558479195378498934987666221630068226125894287176921932312566
circular sector of angle $\pi/3$
14:
233.1173651290127939828581901699426670984103262857560426931111914793498254547127756078494115732905926
equals: #quarter-disk,19
circular sector of angle $\pi/3$
15:
233.6843485098560304593928266858298717699806482277413051432640647870221252208774168954646454982382402
circular sector of angle $\pi/3$
16:
271.2816542728733332513553846235040608959253560036696243725110268597442545280377166495092006753597764
circular sector of angle $\pi/3$
17:
289.6914043115650271802615257457110406970823527826568093557019624656372962427237515870151801593316092
circular sector of angle $\pi/3$
18:
316.4395506424844053598898982853527317856600050936369690222244433303178400142155024736668221779036227
circular sector of angle $\pi/3$
19:
347.3542819385555941315852758981695372902434285374663935432119942154002935972728992787529236164243879
circular sector of angle $\pi/3$
20:
351.3784334796379570542643687592121111014120945961809749226649670191371741416324882957985155952727795
equilateral triangle
1:
0
equals: #unit-square,1
equilateral triangle
2:
17.54596337971441532237242844422426868500213227953918333584595444661341301763414265422537493994409286
equilateral triangle
3:
17.54596337971441532237242844422426868500213227953918333584595444661341301763414265422537493994409286
equals: #equilateral-triangle,2
equilateral triangle
4:
52.63789013914324596711728533267280605500639683861755000753786333984023905290242796267612481983227857
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#equilateral-triangle,1
equilateral triangle
5:
70.18385351885766128948971377689707474000852911815673334338381778645365207053657061690149975977637143
equilateral triangle
6:
70.18385351885766128948971377689707474000852911815673334338381778645365207053657061690149975977637143
equals: #equilateral-triangle,5
equilateral triangle
7:
122.8217436580009072566069991095698807950149259567742833509216811262938911234389985795776245796086500
equilateral triangle
8:
122.8217436580009072566069991095698807950149259567742833509216811262938911234389985795776245796086500
equals: #equilateral-triangle,7
equilateral triangle
9:
157.9136704174297379013518559980184181650191905158526500226135900195207171587072838880283744594968357
equals: #unit-square,16
equilateral triangle
10:
157.9136704174297379013518559980184181650191905158526500226135900195207171587072838880283744594968357
equals: #unit-square,16
equilateral triangle
11:
210.5515605565729838684691413306912242200255873544702000301514533593609562116097118507044992793291143
equilateral triangle
12:
228.0975239362873991908415697749154929050277196340093833659974078059743692292438545049298742192732071
equilateral triangle
13:
228.0975239362873991908415697749154929050277196340093833659974078059743692292438545049298742192732071
equals: #equilateral-triangle,12
equilateral triangle
14:
280.7354140754306451579588551075882989600341164726269333735352711458146082821462824676059990391054857
equilateral triangle
15:
280.7354140754306451579588551075882989600341164726269333735352711458146082821462824676059990391054857
equals: #equilateral-triangle,14
equilateral triangle
16:
333.3733042145738911250761404402611050150405133112444833810731344856548473350487104302821238589377643
equilateral triangle
17:
333.3733042145738911250761404402611050150405133112444833810731344856548473350487104302821238589377643
equals: #equilateral-triangle,16
equilateral triangle
18:
368.4652309740027217698209973287096423850447778703228500527650433788816733703169957387328737388259500
equilateral triangle
19:
368.4652309740027217698209973287096423850447778703228500527650433788816733703169957387328737388259500
equals: #equilateral-triangle,18
equilateral triangle
20:
438.6490844928603830593107111056067171250533069884795833961488611653353254408535663556343734986023214
right-isosceles triangle
1:
0
equals: #unit-square,1
right-isosceles triangle
2:
9.869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
equals: #unit-square,2
right-isosceles triangle
3:
19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
equals: #unit-square,4
right-isosceles triangle
4:
39.47841760435743447533796399950460454125479762896316250565339750488017928967682097200709361487420893
equals: #unit-square,5
right-isosceles triangle
5:
49.34802200544679309417245499938075567656849703620395313206674688110022411209602621500886701859276116
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#right-isosceles-triangle,1
right-isosceles triangle
6:
78.95683520871486895067592799900920908250959525792632501130679500976035857935364194401418722974841785
equals: #unit-square,9
right-isosceles triangle
7:
88.82643960980422756951041899888536021782329466516711563772014438598040340177284718701596063346697009
equals: #unit-square,10
right-isosceles triangle
8:
98.69604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232
equals: #unit-square,12
right-isosceles triangle
9:
128.3048572141616620448483829983899647590780922941302781433735418908605826914496681590230542483411790
equals: #unit-square,14
right-isosceles triangle
10:
157.9136704174297379013518559980184181650191905158526500226135900195207171587072838880283744594968357
equals: #unit-square,16
right-isosceles triangle
11:
167.7832748185190965201863469978945693003328899230934406490269393957407619811264891310301478632153879
equals: #unit-square,18
right-isosceles triangle
12:
177.6528792196084551390208379977707204356465893303342312754402887719608068035456943740319212669339402
equals: #unit-square,20
right-isosceles triangle
13:
197.3920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446
right-isosceles triangle
14:
246.7401100272339654708622749969037783828424851810197656603337344055011205604801310750443350929638058
right-isosceles triangle
15:
246.7401100272339654708622749969037783828424851810197656603337344055011205604801310750443350929638058
equals: #right-isosceles-triangle,14
right-isosceles triangle
16:
256.6097144283233240896967659967799295181561845882605562867470837817211653828993363180461084966823580
right-isosceles triangle
17:
286.2185276315913999462002389964083829240972828099829281659871319103812998501569520470514287078380147
right-isosceles triangle
18:
315.8273408348594758027037119960368363300383810317053000452271800390414343174145677760567489189936714
right-isosceles triangle
19:
335.5665496370381930403726939957891386006657798461868812980538787914815239622529782620602957264307759
right-isosceles triangle
20:
355.3057584392169102780416759955414408712931786606684625508805775439216136070913887480638425338678803
hemiequilateral triangle
1:
0
equals: #unit-square,1
hemiequilateral triangle
2:
17.54596337971441532237242844422426868500213227953918333584595444661341301763414265422537493994409286
equals: #equilateral-triangle,2
hemiequilateral triangle
3:
52.63789013914324596711728533267280605500639683861755000753786333984023905290242796267612481983227857
equals: #equilateral-triangle,4
hemiequilateral triangle
4:
70.18385351885766128948971377689707474000852911815673334338381778645365207053657061690149975977637143
equals: #equilateral-triangle,5
hemiequilateral triangle
5:
122.8217436580009072566069991095698807950149259567742833509216811262938911234389985795776245796086500
equals: Dirichlet_eigenvalues_of_the_classical_planar_domains#hemiequilateral-triangle,1
hemiequilateral triangle
6:
157.9136704174297379013518559980184181650191905158526500226135900195207171587072838880283744594968357
equals: #unit-square,16
hemiequilateral triangle
7:
210.5515605565729838684691413306912242200255873544702000301514533593609562116097118507044992793291143
equals: #equilateral-triangle,11
hemiequilateral triangle
8:
228.0975239362873991908415697749154929050277196340093833659974078059743692292438545049298742192732071
equals: #equilateral-triangle,12
hemiequilateral triangle
9:
280.7354140754306451579588551075882989600341164726269333735352711458146082821462824676059990391054857
equals: #equilateral-triangle,14
hemiequilateral triangle
10:
333.3733042145738911250761404402611050150405133112444833810731344856548473350487104302821238589377643
equals: #equilateral-triangle,16
hemiequilateral triangle
11:
368.4652309740027217698209973287096423850447778703228500527650433788816733703169957387328737388259500
equals: #equilateral-triangle,18
hemiequilateral triangle
12:
438.6490844928603830593107111056067171250533069884795833961488611653353254408535663556343734986023214
equals: #equilateral-triangle,20
hemiequilateral triangle
13:
473.7410112522892137040555679940552544950575715475579500678407700585621514761218516640851233784905071
hemiequilateral triangle
14:
491.2869746320036290264279964382795231800597038270971334036867245051755644937559943183104983184346000
hemiequilateral triangle
15:
543.9248647711468749935452817709523292350661006657146834112245878450158035466584222809866231382668786
hemiequilateral triangle
16:
631.6546816697189516054074239920736726600767620634106000904543600780828686348291355521134978379873428
hemiequilateral triangle
17:
649.2006450494333669277798524362979413450788943429497834263003145246962816524632782063388727779314357
hemiequilateral triangle
18:
684.2925718088621975725247093247464787150831589020281500979922234179231076877315635147896226578196214
hemiequilateral triangle
19:
754.4764253277198588620144231016435534550916880201848834413760412043767597582681341316911224175959928
hemiequilateral triangle
20:
842.2062422262919354738765653227648968801023494178808001206058134374438248464388474028179971173164571
Definition
The Neumann eigenvalues $\mu_n(\Omega)$ satisfy $-\Delta u=\mu u$ in the planar domain $\Omega$ and $\partial_\nu u=0$ on its boundary, where $\nu$ is the outward unit normal [2].
Parameters
$\Omega$
—   domain ($\Omega$ is one of the listed classical planar domains.)
$n$
—   eigenvalue index ($n\geq1$)
Formulas
(1)
For the rectangle $[0,a]\times[0,b]$, the spectrum is the sorted multiset $\{\pi^2(j^2/a^2+k^2/b^2):j,k\in\mathbb Z_{\geq0}\}$.
(2)
For the unit disk, the spectrum consists of $0$ and $\rho_{m,k}^2$, where $m\geq0$, $k\geq1$ and $\rho_{m,k}$ is the $k$th strictly positive zero of $J_m^{\prime}$. Each radial mode has multiplicity one for $m=0$ and two for $m\geq1$ [2].
(3)
For a unit-radius sector of angle $\alpha$, the spectrum consists of $0$ and $\rho_{m,k}^2$ with $m=\ell\pi/\alpha$, $\ell\in\mathbb Z_{\geq0}$ and $k\geq1$. Each pair $(\ell,k)$ contributes one mode, with angular factor $\cos(\ell\pi\theta/\alpha)$.
(4)
For the unit-side equilateral triangle, the spectrum is the sorted multiset $\{\frac{16\pi^2}{9}(j^2+jk+k^2):j,k\in\mathbb Z_{\geq0}\}$ [1].
(5)
For the right-isosceles triangle with legs $1$, the spectrum is the sorted multiset $\{\pi^2(j^2+k^2):j,k\in\mathbb Z_{\geq0},\ k\leq j\}$. Its eigenfunctions extend evenly across the hypotenuse to Neumann eigenfunctions of the unit square [3].
(6)
For the hemiequilateral triangle with hypotenuse $1$, the spectrum is the sorted multiset $\{\frac{16\pi^2}{9}(j^2+jk+k^2):j,k\in\mathbb Z_{\geq0},\ k\leq j\}$. These are the equilateral triangle modes symmetric across the bisecting altitude [1].
Comments
(7)
The domains use the same sizes as the Dirichlet table. The square is $[0,1]^2$ and the rectangle is $[0,1]\times[0,2]$. The disk and sectors have radius $1$; the half disk, quarter disk and 60-degree sector have angles $\pi$, $\pi/2$ and $\pi/3$, respectively.
(8)
The equilateral triangle has vertices $(0,0)$, $(1,0)$ and $(1/2,\sqrt3/2)$. The right-isosceles triangle has vertices $(0,0)$, $(1,0)$ and $(0,1)$. The hemiequilateral triangle is the half of the unit-side equilateral triangle with vertices $(0,0)$, $(1/2,0)$ and $(1/2,\sqrt3/2)$.
(9)
The eigenvalues are listed in nondecreasing order and repeated according to multiplicity. The constant eigenfunction gives the simple eigenvalue $\mu_1=0$ on every domain. Positive eigenvalues begin at $n=2$. The stored quantity is the eigenvalue of $-\Delta$, rather than its square root.
(10)
Write $\rho_{m,k}$ for the $k$th strictly positive zero of $J_m^{\prime}$. In the Bessel-extrema table, the zero at the origin is included for order $0$, as in [4]; hence $\rho_{0,k}$ is its entry $(0,k+1)$, while $\rho_{m,k}$ is its entry $(m,k)$ for $m\geq1$.
Programs
(P1)
Python
import mpmath as mp
mp.mp.dps = 120
# First positive Neumann eigenvalues; n=1 is exactly zero.
print("square, n=2:", mp.nstr(mp.pi**2, 100))
print("rectangle, n=2:", mp.nstr(mp.pi**2 / 4, 100))
print("disk, n=2:", mp.nstr(mp.besseljzero(1, 1, derivative=1)**2, 100))
print("equilateral triangle, n=2:", mp.nstr(16 * mp.pi**2 / 9, 100))
# mpmath includes zero in order zero: index 2 is its first positive root.
print("first positive radial disk mode:",
      mp.nstr(mp.besseljzero(0, 2, derivative=1)**2, 100))
References
[1]
R. S. Laugesen, Z. C. Pan and S. S. Son, Neumann eigenvalue sums on triangles are (mostly) minimal for equilaterals, Mathematical Inequalities & Applications 15 (2012), 381–394. (doi)
[2]
Jonathan Rohleder, Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains, Mathematische Annalen 392 (2025), 5553–5571. (doi)
[3]
Ram Band, Michael Bersudsky and David Fajman, A note on Courant sharp eigenvalues of the Neumann right-angled isosceles triangle, 2015, Section 2. (arXiv)
Links
Similar tables
Dirichlet_eigenvalues_of_the_classical_planar_domains —   the same domain geometries with zero boundary values instead of zero normal derivatives
Local_extrema_of_Bessel_J_functions —   squaring the positive extrema locations gives the disk and sector eigenvalues
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds the first 20 eigenvalues, including the zero mode and multiplicities, for the square, rectangle, disk, three circular sectors and three triangles listed here. These low modes provide reference values for membrane spectra and numerical eigenvalue solvers. The annulus, L-shape, two Gordon–Webb–Wolpert drums and regular polygons from the Dirichlet table are omitted)
How they were obtained:

Nonzero entries are stored to 100 significant digits. Polygonal spectra are enumerated as exact rational multiples of $\pi^2$, with $\pi$ evaluated in Arb ball arithmetic. All zero modes are stored as exact integers.

more

For the disk and sectors, SciPy selects candidate mode indices. Each selected Bessel derivative root is recomputed with mpmath at 130 and 160 decimal digits, and the computations must agree to at least 120 decimal places. Arb evaluates $J_m^{\prime}$ at exact rational endpoints around each root and certifies opposite signs; squaring this bracket gives an enclosure of an eigenvalue. The overall rigour remains agreement-checked because identification and enumeration of the root indices are not certified by an interval root-counting algorithm.

The attached independent checks compare all 40 equilateral and hemiequilateral entries against the mode indices in [1], Table 2; check all rectangle and right-isosceles entries using cosine energy integrals, beginning with a known nonzero control; and verify the right triangle boundary conditions. They independently enumerate the disk and sector modes with mpmath and check the actual written decimal endpoints with $xJ_m^{\prime}(x)=mJ_m(x)-xJ_{m+1}(x)$. The 72 circular entries within the coverage of the Bessel-extrema table are also compared with that table.



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