Characteristic values $a_n(q)$ of the Mathieu equation

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Numbers

$q$

$n$

$a_n(q)$

-0.4551386041074135482326331875288858669165198421966654710317486675822261235164160470159425623418250589

1.859108072514363472329917512841423300127410341317695081625775769043966104074314694144409579202762489

4.371300982735085661171120918199422244868492771912378157766245915462346356167914760113086656176209254

9.078368847203101992542227389838886773846035067565821921475351184485590771532228511305502042807537785

16.03383234035951396115310292247387218351311829540466158126228671529568438517272883259055193928865027

25.02085434544858105321885332923390473871795259780880798928037652340677700614002341588557773473264703

36.01429004604049922203489434681190541280766028060644661189175650238832013276422730013130353709670056

49.01041825036485279618077257200961372365694752070295449051413140270824516896976883381858925789446006

64.00793718925467418869817603308066535182183622326957198192325631647331426322701336074013750443234299

81.00625032663259684324348148794582909327212528430737639890439786173206877812709196249076067328513169

10:

100.0050506751594644392114505162757105040259099215869168613957919988409052483841699338935298735106845

1/2

-0.1217655449410826959442311040306376080721390230870514597566306137582620584666945364398653581765744783

1/2

1.466766842516055774308238490098876885672560272473706374801003835403979645481582209215219603032449048

1/2

4.100900595560480626440489824331561294060149376829491562485697274596914740203961592175034639444236598

1/2

9.017606927797508666664672353869651055518659076593238410461923888085305282370989282757671476733208647

1/2

16.00836462272300738824811359125159844797828806634741006833461607254723097522414026718343797181198691

1/2

25.00520943382713712344600382449307492019434013603038899438761993078037365668246726417982741223002943

1/2

36.00357169591063031522519472098600157826951290061620214287557145119814920299897428595007208469172654

1/2

49.00260426558293770570452311256412501982966133175699699667414622316508313294605670985440301641832012

1/2

64.00198416955728547894957882632918159284841419964743794961426865223166697116552114809355514010631432

1/2

81.00156252041196460903368770896855035525739007695094904942107875428041959401138305353594010065502937

1/2

10:

100.0012626368935912164629661292596375992217717242222594399942103301654472828506854924219377716681311

-1.513956885056520254181643952263881021949068787817187068660392445849420212558570792864098704258150306

2.379199880488686029803809752454532306373741290498460568853094811701863325584713744478579045495258950

5.172665133358294193203854859202831839908089575366441002779336192962730845247857265703409480229956989

9.370322483621104012944711601480567278968253843604621840262910116867044296749375724411199010262582904

16.14120378558493654182861394362272251930212095099842290054792991854211602243066576162218168069652635

25.08377778277331010228825984598053601350574832869307972155372554907983626763257478516319348624729283

36.05721562419987781348859745443850957145990206399936000870896867980649721706596107302377636781968705

49.04169209857535048543676175418884184373819242787653950619079656318296462281106688497226741603228755

64.03175694273274356144049246757719621526795541503011938691430126934561730039657272096614010146212784

81.02500522876272709810731686721618815944972842409414519780737787872027955401579252590283205892108709

10:

100.0202047428112256104570604168341337250735629702311511894310050540701818700544903396687563831079711

Definition

For real $q$, $a_n(q)$ is the value of $a$ for which Mathieu's equation $w''+(a-2q\cos 2z)w=0$ has the even periodic Mathieu solution $\mathrm{ce}_n(z,q)$, with $a_n(0)=n^2$ and with the branch of $a$ followed continuously in $q$ from $q=0$ [1].

Parameters

$q$

— Mathieu parameter ($q>0$)

$n$

— characteristic index (integer with $n\geq0$)

Formulas

(1)

$a_n(0)=n^2$.

(2)

$a_{2m}(-q)=a_{2m}(q)$.

Comments

(3)

The normalisation is DLMF's [1]: the oscillatory term is $2q\cos 2z$. Conventions using $2q\cos z$, $q\cos 2z$ or $16q\cos 2z$ give different values when $q\ne0$.

(4)

The companion characteristic values $b_n(q)$ belong to the odd Mathieu solutions $\mathrm{se}_n(z,q)$ and are not stored in this table. They give the remaining reflection, $a_{2m+1}(-q)=b_{2m+1}(q)$, and they interlace with the values here: $a_0(q)<b_1(q)<a_1(q)<\cdots$ for $q>0$.

(5)

Together with the companion curves $b_n(q)$, the curves $a_n(q)$ form the boundaries of the stability regions in the Ince-Strutt diagram [1].

(6)

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.

(7)

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].

Programs

(P1)

Python

from scipy import special

# a_4(1) in DLMF's normalisation
special.mathieu_a(4, 1.0)

Links

[1]

DLMF Chapter 28: Mathieu Functions and Hill's Equation

[2]

DLMF 28.6(i): expansions for small q

[3]

SciPy reference: scipy.special.mathieu_a

[4]

DLMF 28.3: NIST reference plots at q=1

[5]

DLMF 28.2: Mathieu parity and periodicity

[6]

DLMF 28.4: Fourier recurrence and normalization

Similar tables

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Eigenvalues of the pure quartic, sextic, octic and decic oscillators $-y''+x^{2m}y$ — also stores eigenvalues of one-dimensional Schrödinger operators, for polynomial potentials rather than the periodic potential $2q\cos 2z$

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds $a_n(q)$ for $0\leq n\leq10$ at $q=1/2,1,2$, the three coupling slices in (7). Companion $b_n$ values are not part of this table)

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.

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.