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Values of the even Mathieu functions $\mathrm{ce}_n(z,q)$

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Numbers
$n$
$q$
$z/\pi$ 
$\mathrm{ce}_n(z,q)$
0
1
0:
0.3848278293012990758474910248336609810122001980980129771942540534965181322934317893785426065085512022
0
1
1/2:
0.9984585148130366754367663964161154017571655255749563294584260830674955464046996460635322546691075099
0
1
1/4:
0.6543520522319160858435793356487624964423617603987642136456122111227297690793062370077562465846557461
0
1/2
0:
0.5323312580565888177231559820461272976842710575964035349911349382189143993617496672025465670598327884
0
1/2
1/4:
0.6915756732700816908030675975546420989905171509688705393232345427111615538837557565686523661368327289
0
1/2
1/2:
0.8718817048959044885782208808388145559484440962409562776354179142414034053424289665111183840568434123
0
2
0:
0.2026172657999473554470463539037360296334783184440757110399447919415255125437625122318981836165398645
0
2
1/4:
0.5684475796316198635779130165639732673031077014389538468274026220370628968084247509835940674190162712
0
2
1/2:
1.150841427475981021353893553841965127143527267550611640223988898728834484877301387509523332993080709
1
1
0:
0.8565984655568863879714013816997302525844457432767607382129353601207054413699380816023324856144169994
1
1
1/2:
0
1
1
1/4:
0.7944718100100970473072219645520738435212867851434276428592750475327383506464543108727749274097567062
1
1/2
0:
0.9328721490152929830622373548025074907440023779065332918473422248674239108812010682608152932309433232
1
1/2
1/4:
0.7514375790341728351048257028201824871841171304443594203874303309485551785059180865588541302235716123
1
2
0:
0.6835733015409525591525740709625407772651006576359902463470959054204769333442649363273353071425106192
1
2
1/4:
0.8652593921093095951780802814860751076057974369033953404754558425118708629650891369305934616479148435
2
1
0:
1.085961936889011546924767385620378072011519693447782476698897943964123549129497966910950761275966996
2
1
1/2:
-0.8157268390500981910410364475835156048612374738345507651348840568723567596284437793169191783646444570
2
1
1/4:
0.2986515729070593070788373382666143889016403550594398795827790500483269123401919831671879924935311628
2
1/2
0:
1.063904088741245843213018670026751779245179034390090639893217143229882720192147006871513453078070687
2
1/2
1/4:
0.1614436311832794951249225703613558005643508027452907176970477335680736559978249116780036756733098137
2
1/2
1/2:
-0.9066195790837282204192518596079760872394933434719188896580837419159595419230320517970114532501923207
2
2
0:
1.048834490252726293849194561122936974843617037505426209384511859019486077716704716490018610428491885
2
2
1/4:
0.4962617092569275404992874264329178979524997231909798598019954256668445604315796437669043019106474444
2
2
1/2:
-0.7037857603559417798024348060468584338695109074583026108086391882675273940171043654955250501590793779
3
1
0:
1.067235512293139057634005081113604434115619042284305661494052349771482143677247099559508070673225380
3
1
1/2:
0
3
1
1/4:
-0.5550307133804575022249931941237445235659720361308683126845017180523190491780224135441435020065926750
3
1/2
0:
1.032816022799428204105430750160695058539025463608846166214786906197043705870026899655584789879183840
3
1/2
1/4:
-0.6359559956743257030471154418702275336780720851724799683099675718347440912034731064036559624358800665
3
2
0:
1.128274810917938330713711260594374298380774966849726897309517481352618086119098018096417651426020506
3
2
1/4:
-0.3688941379889492969767626381303995467331803913351018826693911166541164723831975894541937881616239908
4
1
0:
1.035135158578013299358880261996245092179989550838456530416433246153553242642725047543447778695745992
4
1
1/2:
0.9678184211483232852243800907471627886512364432243628403491951919055736716901345141666567807451266179
4
1
1/4:
-0.9889717294331822227140214206981985072585005716435157027071614097441007018443354920458777244206295553
4
1/2
0:
1.017086659582120471347584763511074483315028341409813156535538933591889524136911600409390863498028364
4
1/2
1/4:
-0.9972533888620349304898173173379955386218620054438111798289820539862082800667378300189242254935707034
4
1/2
1/2:
0.9836708808622902897175821645789966441364555390546718298187109575899600340505841776878201513371248895
4
2
0:
1.074395048823757534169844551966569910345049582225092484024764043780718313171195811840812265238041594
4
2
1/4:
-0.9552704463679352270795045300141482534447419597361687053757084132590048206286077218510858175690459741
4
2
1/2:
0.9361742941511047233827518441065947099190557670504107293694619606600754666531075571068314439724100398
5
1
0:
1.021420968168026927718090850744496800066685337240584406376150984932155365950329602171087119036779786
5
1
1/2:
0
5
1
1/4:
-0.7762223297572337371818789755429631860944215538351101468205141983540956680581124376062431558778076368
5
1/2
0:
1.010555745207132011334805527862865032303591747235795571827564035681877880362072516501307312517146172
5
1/2
1/4:
-0.7428219290692485325480511846447595394305200125055430404552817311420934308096814558063527349960458117
5
2
0:
1.044282999413490861256659236875752524326039482357499728114803945380180394813484640956202889866287980
5
2
1/4:
-0.8355745776302206667406396451212158665287722578298407372748208949888882516217277992269860758292782776
Definition
Listed are values of the DLMF even Mathieu function $\mathrm{ce}_n(z,q)$ [1]: the even periodic solution of $w''+(a_n(q)-2q\cos(2z))w=0$, where $a_n(q)$ is the $n$th characteristic value in DLMF's $a_n$ sequence. The normalization is (2).
Parameters
$n$
—   order ($n\geq0$)
$q$
—   Mathieu parameter ($q>0$)
$z/\pi$
—   argument divided by $\pi$ ($0\leq z/\pi\leq1/2$)
Formulas
(1)
$\mathrm{ce}_n(z,q)$ satisfies $w''+(a_n(q)-2q\cos(2z))w=0$.
(2)
$\int_0^{2\pi}\mathrm{ce}_n(z,q)^2\,dz=\pi$.
(3)
$\mathrm{ce}_0(z,0)=1/\sqrt2$, and $\mathrm{ce}_n(z,0)=\cos(nz)$ for $n\geq1$.
(4)
With DLMF Fourier coefficients $A_m^n(q)$ [2], $\mathrm{ce}_{2r}(z,q)=\sum_{m\geq0}A_{2m}^{(2r)}(q)\cos(2mz)$ and $\mathrm{ce}_{2r+1}(z,q)=\sum_{m\geq0}A_{2m+1}^{(2r+1)}(q)\cos((2m+1)z)$.
Comments
(5)
The argument $z$ is measured in radians. A row with $z/\pi=\rho$ stores $\mathrm{ce}_n(\rho\pi,q)$; software that takes the argument in degrees must convert before comparing with this table.
(6)
The order and sign are fixed by continuity in $q$: $\mathrm{ce}_n(z,q)\to\mathrm{ce}_n(z,0)$ as $q\to0^+$, with $\mathrm{ce}_n(z,0)$ as in (3).
(7)
The values on $0\leq z\leq\pi/2$ determine the function on the real line: $\mathrm{ce}_n(-z,q)=\mathrm{ce}_n(z,q)$ and $\mathrm{ce}_n(z+\pi,q)=(-1)^n\mathrm{ce}_n(z,q)$.
(8)
The coupling $q=1$ is a modest reference case used in the NIST plots [4], in the normalization $w^{\prime\prime}+(a-2q\cos2z)w=0$. It is not a uniquely distinguished physical coupling. The integer order labels a spectral branch; the upper order bound is a finite reference depth.
(9)
Parity and periodicity make the endpoints $z=0,\pi/2$ zeros or stationary points, depending on the branch. Exact symmetry zeros are stored as $0$ [5].
(10)
The couplings $q=1/2,1,2$ form three bounded exploratory references for the Fourier coupling in $w''+(a-2q\cos2z)w=0$, not uniquely preferred physical constants. The integer order is a spectral index; its finite depth is a reference choice [6].
(11)
The argument $z=\pi/4$ is halfway between the reflection boundaries and has $\cos(2z)=0$. It supplies nontrivial interior values for branches whose boundary values vanish; it is not generally a stationary point of the eigenfunction.
Programs
(P1)
Python
from generate import MathieuReference

generator = MathieuReference("T437")
generator.value({"n": "1", "q": "1", "z_over_pi": "1/4"}, 100)
Links
Similar tables
Values of Bessel functions of the first kind $J_\nu(x)$ —   stores values of another classical special function used in separated-coordinate eigenfunction expansions
Values of the Airy function of the first kind $\operatorname{Ai}(x)$ —   stores values of another classical special function defined by a second-order differential equation
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds $0\leq n\leq5$ at $q=1/2,1,2$, using the symmetry boundaries $z=0,\pi/2$ and the interior point $z=\pi/4$ where the oscillatory potential vanishes. Existing exact zeros at $q=1$ are preserved; additional identically zero boundary rows at the other two couplings are omitted. This is a small set of structural argument slices, not an angular mesh)
How they were obtained:

All earlier numeric strings and manual entry metadata are retained. Every value is recomputed using finite Fourier matrices at 160/210 decimal digits and maximum frequencies 110/150, and independently checked by Taylor ODE shooting at 160/190 digits, 12/16 steps and degrees 140/170. The ODE solves the boundary condition and obtains the normalization from the eigenvalue-derivative Wronskian.

more

Known zero-coupling trigonometric solutions and their norms control the independent computation. At the boundaries the Fourier factors are exact parity signs; interior arguments and pi are recomputed at each working precision. Every printed 100-digit interval contains the independent value. These checks establish numerical agreement, not rigorous infinite-operator truncation or ODE remainder bounds. The attached coverage-checks.json records the checks for this selection.



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