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Zeros of prolate spheroidal wave functions
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Numbers
$m$
$S$
$n$
$j$ 
$x_{n,j}(c)$
0
1
2
1:
0.5345062822074797121058232490420341006235381808592859291479480846007905751565429391297424172030519214
0
1
3
1:
0.7570402971559971812457674177192435619933240160992597827412948483185742119972400811002607725907578383
0
1
4
1:
0.3306409268481798117017320211314297599778029342124732522419114220785449352235596021181501550996822854
0
1
4
2:
0.8537582846643680475186662637605231524448478478709270852997525708704212699582515584173508569811110549
0
1
5
1:
0.5304762157659595036568281697587117053528367278131315461253457417635320658820014304299226124230347824
0
1
5
2:
0.9026679968818836364655937964186980904561144433383110212380848712755772611696579361776397445338025884
0
1
6
1:
0.2353095446342794682114375167042570540863102482167224900317974789824262273648903809343436251845236391
0
1
6
2:
0.6556403274224882995270446373960346545381651204595768295770675371048114013501756486858981266420877285
0
1
6
3:
0.9306107351814281316655192558497993846519427279588864564111558443430365209273928162364767389457972572
0
1
7
1:
0.4020930761500969130725512332197151805044214414819840269459202397568931435248442252841009065040736466
0
1
7
2:
0.7377877951456735094618002494467865176701594635299314672559177224479982067827067701866997139445109113
0
1
7
3:
0.9480390173282785839798672830227888693336739066452585141619693912700775834759809740397250126228052718
0
1
8
1:
0.1819154244567644469906922131884539343733619733918066644071471667324591476802378025784416235835806104
0
1
8
2:
0.5222515686433225656587305237574921522759189541038524676645491406196725082016403704260497949129094709
0
1
8
3:
0.7941322811560330901336411202913592404525993398231627943128126590358449252177449972099522667244015683
0
1
8
4:
0.9596339000140387024171105287583614507743650391685720289073560964641870792873562085767836133968800178
0
1
9
1:
0.3222610898932811152725250509536358508252048803968367167873144333964215042666208719848963299259782423
0
1
9
2:
0.6107294101153395781182505200787431908329715688816625771869942638609222976911146499503699453539432793
0
1
9
3:
0.8342812092005116225464759065949435707247363871794726618957760618416714698871934085697618940169874841
0
1
9
4:
0.9677362653030982052179472481667927898712385775275801167206846070866237632427259499531058790747135553
0
1
10
1:
0.1480579219165974755640555965447426634474793396662314393932590307854835787489214865756394726790453836
0
1
10
2:
0.4314157884375458258525652496243473217429924440596419760255836371853803032191694045452438300571416821
0
1
10
3:
0.6773426636035910217472624470852592971558698225752902074474627773768784727642327538745534544694235024
0
1
10
4:
0.8638271317890562607423659046092742598019644433607241410267979210507874902931426381973923687818776227
0
1
10
5:
0.9736206930617377777544283198141708448351145145448977560285846642132791641689457083588853095349837952
0
1
11
1:
0.2683741312119326561082785324692597475478801215683088571826226502605730623546729623851177865721714820
0
1
11
2:
0.5173176256039678826817586623222746868062684434512608766794062977141607070879304775617229937825882597
0
1
11
3:
0.7285472433089918656775850034834445621782949279295094451222531459886375155454309725769868882027770613
0
1
11
4:
0.8861693931061921319994394916691747584898299899863429520492298355544973585596903855194077457371910018
0
1
11
5:
0.9780291135537550466863783981390870056349957271840456580452195259037889319469847743618336536272752869
0
2
2
1:
0.4387487099397048950827642293478765679490645372018176034356660502252726619836663547236123495086735206
0
2
3
1:
0.7000613498979598649676485628600836183479098600764845273347427710586758868067810200710436712178144668
0
2
4
1:
0.3057219111381197839270909755736595771492782061407285511223638239823968185440542976412010972644514518
0
2
4
2:
0.8285822484561063122889245769331460133704572986915793929707564141073413259016163276683053359976110434
0
2
5
1:
0.5073891546524178126113379724967388203898976159402704517274394686208857829570724865970378657882204822
0
2
5
2:
0.8909143246117756515846346811000108668779011543585220109752526952077618823577567115634651259100486314
0
2
6
1:
0.2260008358130312755357015836551831252980409208554120113570888179610425719719007951015331078955558285
0
2
6
2:
0.6389280505103333666133406660410580530319077119194761170785403516600283362848974733288246228717645477
0
2
6
3:
0.9245314840025560213768419733596275956681655529887248898222082241553439320162207496279183939975866675
0
2
7
1:
0.3912218216031565641962343504693664585585605069735784829983154049212112231206288459866005422159495279
0
2
7
2:
0.7263717714122594981335471764712692688544335576365414404682795975922915516489658444284039557472195218
0
2
7
3:
0.9446058878064068910468570558126517515677062647424338221773687880231750134312132810033257034439896038
0
2
8
1:
0.1775350936267594409292525130224595944069118397811020303794409736714759432990574132270173958073457845
0
2
8
2:
0.5125616044540755728269502648265322795572112735316988366544380329675667213054650658788934457606020977
0
2
8
3:
0.7863590294589507021515930765125051827948506474971752875615251684945124470816054718016937712270327801
0
2
8
4:
0.9575552722235250499947196832562705652800468083962525511315965016580598407878114079946846186683540988
0
2
9
1:
0.3164378790098401037093957351615436701193427646297357312437274573523733364304949408741790366165511886
0
2
9
2:
0.6028356664018765650450063272936085128246996718508159954623661376881704254458359675455770778543099266
0
2
9
3:
0.8289072563312852736895334047571322352377930816263204443546303040660864087366060745859939367006499124
0
2
9
4:
0.9664061892628369045401381908970785634745551497822289001110219451504781555930594633231442651344965607
0
2
10
1:
0.1456731074265346297597882386033012153500842311274140221301894411999042905108209465967021796430168693
0
2
10
2:
0.4255702477608024604038601736489987017214847151082753815146433545247431781974097695296198728648553342
0
2
10
3:
0.6711258877461377979588535897072164517262511386115928664307797239556044676026696998999518677769944974
0
2
10
4:
0.8600335682938656496170271852312611637282368979212294607608733110304238935786240789930465229169780549
0
2
10
5:
0.9727307785750677243621943640711971895744765024928042505713194056863449373183665427502710577929846810
0
2
11
1:
0.2649346339192195137356937097308034406776655939534432756319074635232515262218068179383543206654851351
0
2
11
2:
0.5120278260121374449035072869082693528005814013176382284600660680454660323426747799744278786753638669
0
2
11
3:
0.7237025000721549278026204741631990739693500439591479193074896707700405859288723150005987019821824745
0
2
11
4:
0.8834326818333377672056251244990845062369959674176816669788987915107285485173203409940485583554613768
0
2
11
5:
0.9774114893182289142159437843168146980192675285082379143071192629398455865133176206555755794563630956
0
4
2
1:
0.2984173906816629881829921197961258120878792257491394716308537064672841087564127739883314238808492146
0
4
3
1:
0.5234565229913070430462064344950416174173595623469685868363411143677121020467538961573636803930060041
0
4
4
1:
0.2369228226086689910871035083440655066190163623852409182163940280482730656568817196057060407836612236
0
4
4
2:
0.7015650993227050424009973343281034195833331697065907301161781390215682022513078689623416608840393611
0
4
5
1:
0.4287494821294446087964864068090910758090588193076517443783635407356570578551445882340791528147847566
0
4
5
2:
0.8227671616936789539455969984489481230622440449752157618918565821284661338531587408054704607004811997
0
4
6
1:
0.1959075828662862558315798142784325010979179800445483982929127723279638756169255989191387056226647262
0
4
6
2:
0.5741756803682494592228046703878011107722876801456827732461588434890595855371691916637157216451650732
0
4
6
3:
0.8904376815090457725644173476415701146020072930036811083850897193699547778772855675608418801761632642
0
4
7
1:
0.3529580918567328837700315302133166083769427523294439604877820997167057848913264559197423451723258034
0
4
7
2:
0.6789170622418867776876931221679367425986333525968770373457236179267437479190490363453041108786742659
0
4
7
3:
0.9265433913989581139966694181982118419586474544997462373345760213648888004536627432269410917142374413
0
4
8
1:
0.1622674173930204095777906649996836638827443088231918674382959131424430496925615414309890792627351500
0
4
8
2:
0.4762296024331642396429283292279771283189544004775521699389871896567430543027784425716081400550300213
0
4
8
3:
0.7529882164817476632430915551662260895739760604909379446077290172561417062863485562417839110006522464
0
4
8
4:
0.9471995027293993316792504233590626769960070653819201300653736066755519154876052516896868089623889456
0
4
9
1:
0.2952469403929655270202770691320272436474788500153430381141180190255584464817968416709170988544297162
0
4
9
2:
0.5719606263967984171457126791576406932331637536138211094164684295572049648261133544159032734798883098
0
4
9
3:
0.8055801853436686837994385118193649714540789097261495203315249791532627380630501611534202711517232405
0
4
9
4:
0.9600499324482395852387266677484969964203287836503338464527761650793714006186688740933442024292723083
0
4
10
1:
0.1369966844522780195625818674938953206836655584372874138378888683380803661196398977212657948356151597
0
4
10
2:
0.4035504070051493238874420344119032384463247828712442608000378745714952433865642589197054834821849891
0
4
10
3:
0.6461592589222264825782527160607140886974409157835618780538451561348865058415665881674528478406962797
0
4
10
4:
0.8435563211940579580984934537120716596513306803348636123990770759901675162216996646599168951423308413
0
4
10
5:
0.9686096137110610346141235144791826779796078971328978883150164765893018248318069928673178016294582242
0
4
11
1:
0.2521167129576242495386081661486297068122551495686132625822323178772836520197937475387076476149692781
0
4
11
2:
0.4915822577348223891699955239853488423885740652743943840896899060983768116334238252526928707343746311
0
4
11
3:
0.7039395266731231684533685867984600262561696241432698976971531145611424892775201491653751216967967817
0
4
11
4:
0.8715936254554474475401413186974440894317091088858600066738130515499690641555449365059758248157419203
0
4
11
5:
0.9746191738667202346079359441666175997661679712529318178938515240504825311215605226404559168894923459
0
8
2
1:
0.2040372040173682388098374568579831726003877969727735221250436328141772915143224395750462529879744683
0
8
3
1:
0.3540879740842316223869520570529803969010298138180178410161640139222427531581280244898498454265294591
0
8
4
1:
0.1557779055329149620913767130638180660115145219090545532331790395031368322041539520445300353592983600
0
8
4
2:
0.4773845501025038733151471598820084366352910595070186374162114658959348025761823206466586583925950883
0
8
5
1:
0.2871664398428695731664787098216390491093979047021521392947160572198085625454729183598000682873976667
0
8
5
2:
0.5842748099691095157405113202999976062534636939565571785102661387921798343136268205010600565163892438
0
8
6
1:
0.1341127844223753899109333011945867655831823323798482954814627032954385799092847384368076838006441175
0
8
6
2:
0.4038689893323218666852316241616896335376895333134008830286069919714343279694519310476834788679607645
0
8
6
3:
0.6805008666275072609945960260481260817760708634244350303325302890717260836602267411734752322613587181
0
8
7
1:
0.2551398970401969692956384392892244116983126810519047360505877221042768210042951069631554741811790639
0
8
7
2:
0.5109459782370292067588346948307448336738405124801123717743371999825460986184623076709567402584737657
0
8
7
3:
0.7687224107906222028344762256973837981177501070730000860840666356492816477897683665132754959080219493
0
8
8
1:
0.1221858879840911066209290300355013418304124028216009833519060056967397317649050585313108478637271460
0
8
8
2:
0.3661588780374055545296531293093828845083768672048603676790362952761247769869848042293707651266779312
0
8
8
3:
0.6085203950236877444684500567323566560813116994072445396088069728347722738028043357726991673731516031
0
8
8
4:
0.8450133886600830327850748676834510835063086165185108690394954351275408816838721938270669900115112487
0
8
9
1:
0.2337859006361518167085807014861377366831222234859206463314899733313500774870649955707585477097190507
0
8
9
2:
0.4656634044283315878608847211079644163559147160124412305850272651374125751268548553198174010430431105
0
8
9
3:
0.6920867073735537952730055606819409142686045161715654790508833579065513118024975372755305865850004915
0
8
9
4:
0.9007725920383213929155948486408905459808884856470060130788678831957639131268954676730968083070390214
0
8
10
1:
0.1116330838228143185720175145228631994401585483744135724714724226244009027662204564735228222567760750
0
8
10
2:
0.3336880734730562793230495920585070767104608637067712235753327203064687801638003681280667998983314287
0
8
10
3:
0.5514209286565315221392469667310756109042421813247790551766437625207973570655280387608527747442982105
0
8
10
4:
0.7585061864353688006868493667851427482406817743489595313309811164562624216768117864524408058552181768
0
8
10
5:
0.9348265720027436664704511196943513375585603225982342507675624539519186354455550486898422683225814995
0
8
11
1:
0.2121970332157093046580254443806234385193189238526461700438796179007840530762890611541890442362901670
0
8
11
2:
0.4214371714172533395211538846151031540532974665596553871756134246343117323797909913534025821149857570
0
8
11
3:
0.6233695202087803359527634420525722908565581144543749188217863273765690336400839678478278592372566466
0
8
11
4:
0.8090888011834653629027727946083972097309670949398459589477293663018711590248429140360127803729514694
0
8
11
5:
0.9544256403402115917880686259768895777648950670483381696973120236759197936843621525334093391214561759
Definition
The numbers $x_{n,j}(c)$ are the positive zeros, counted increasingly from $j=1$ in $(0,1)$, of the order-zero angular prolate spheroidal wave function $\psi_n(x;c)$, with $c=\pi S/2$ [1].
Parameters
$m$
—   angular order ($m=0$)
$S$
—   Shannon number ($S>0$)
$n$
—   mode index ($n\geq2$)
$j$
—   positive zero index ($1\leq j\leq\lfloor n/2\rfloor$)
Formulas
(1)
$\psi_n(-x;c)=(-1)^n\psi_n(x;c)$ [2].
(2)
$\psi_n(x;0)=\sqrt{(2n+1)/2}\,P_n(x)$, where $P_n$ is the $n$th Legendre polynomial [2].
Comments
(3)
The function $\psi_n$ is a nonzero solution bounded at $x=\pm1$ of $((1-x^2)\psi_n'(x;c))'+(\chi_n(c)-c^2x^2)\psi_n(x;c)=0$. The differential eigenvalues $\chi_n(c)$ are indexed increasingly from $n=0$ [1].
(4)
The angular order is $m=0$, distinct from the mode index $n$. We choose the real eigenfunction with $\int_{-1}^1\psi_n(x;c)^2\,dx=1$ and $\psi_n(1;c)>0$; multiplication by a nonzero constant does not change its zeros [2].
(5)
This table concerns angular zeros inside $(-1,1)$. Each mode has exactly $n$ simple interior zeros [1]. The negative zeros are the reflections of the positive zeros, and an odd mode also has the exact zero $x=0$. Modes $n=0$ and $n=1$ have no positive interior zeros.
(6)
The Shannon number $S=2c/\pi$ is the trace of the sinc operator whose concentration eigenvalues share these eigenfunctions. It measures effective dimension; the four selected integral values are a finite reference choice, not a distinguished set of all possible bandwidths.
(7)
The interior zeros can be used as nodes in quadrature for bandlimited functions [1]. As $c$ tends to zero, they tend to the zeros of the Legendre polynomials, which are the Gauss–Legendre nodes.
Programs
(P1)
Python
# Double-precision approximation on a bracket containing one positive zero.
from math import pi
from scipy.special import pro_ang1
from scipy.optimize import brentq

def positive_zero(S, n, a, b):
    assert 0 < a < b < 1
    return brentq(lambda x: pro_ang1(0, n, pi*S/2, x)[0], a, b,
                  xtol=1e-15)

positive_zero(1, 2, 0.5, 0.6)  # approximately 0.534506282207
References
[1]
A. Osipov and V. Rokhlin, On the evaluation of prolate spheroidal wave functions and associated quadrature rules (2013), §§3.1–3.2 and 5.3. (arXiv)
Links
Similar tables
Prolate spheroidal differential eigenvalues —   These eigenvalues specify the differential equations whose bounded eigenfunctions have the tabulated zeros.
Slepian concentration eigenvalues —   The continuous concentration operator has the same order-zero eigenfunctions and mode indices.
Power-series coefficients of spheroidal eigenvalues —   These coefficients describe the differential eigenvalues near zero bandwidth.
Nodes and weights of Gauss–Legendre quadrature —   The angular zeros tend to these nodes as the bandwidth tends to zero.
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds every positive interior zero for $m=0$, $2\leq n\leq11$ and $S\in\{1,2,4,8\}$ in the normalization $c=\pi S/2$; these four noncanonical reference bandwidths successively double a small integral effective dimension, while the practical mode cutoff covers low modes of both parities and both sides of the concentration transition at the larger bandwidths)
How they were obtained:

The generator evaluates parity-separated expansions in the orthonormal Legendre basis. Indexed Sturm bisection determines each differential eigenvalue, backward recurrence determines its expansion coefficients, and Newton iteration refines the positive zeros [3]. The attached argument-selection.md explains the bandwidths, mode range and zero indices.

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Working precisions of 160 and 220 decimal digits are compared at 80 terms per parity block. At 220 digits the basis is independently enlarged to 120 and 180 terms. The minimum relative agreement over all 120 zeros was 142 decimal digits; 100 significant digits are retained.

The independent checker solves the differential equation by a Taylor series from $x=1$, normalized by $\psi(1)=1$. It determines its own eigenvalue by imposing the even or odd boundary condition at the origin, then refines the zeros using that solution. Computations at 220 digits with 600 Taylor terms and at 260 digits with 800 terms agreed with the largest Legendre calculation to at least 202 relative digits. Nonzero Legendre values and derivatives, together with the positive zero $1/\sqrt3$ of $P_2$, control the checker. The zero-bandwidth values and derivatives are compared exactly for modes $0\leq n\leq11$.

For every retained decimal, both endpoint computations and the largest Legendre computation take opposite signs at the endpoints of the interval denoted by its last written digit. Each mode has its expected number of disjoint positive intervals, and parity and the odd central zero are checked for every selected mode. SciPy supplies initial guesses and an external double-precision comparison [4]; its largest absolute zero discrepancy was less than $3.4\times10^{-13}$. The attached checks support the same tests on the values read back from the database.

These are numerical agreement checks. They do not establish a rigorous truncation bound for the infinite Legendre expansion or the endpoint Taylor series, so the rigour remains heuristic (agreement-checked).



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