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

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Numbers

$n$

$q$

$z/\pi$

$\mathrm{se}_n(z,q)$

1/2:

1.108009466037005171668883637723350942670449565835476519722068346018174983322197534318245264880339317

1/4:

0.6223293212236893619472104957363078705735205813532347615747279636768174052880404831431565921202210178

1/2

1/4:

0.6636259431266465690315706354936906623363751653311090835078263248116248356393377487854186334605126132

1/2

1/2:

1.058078741833098244313645050313963375254432844788830077674005893890256912926991833931814437276534086

1/4:

0.5484838095553090155131589122147806276510043420022617870981167848326842713827142601476947570955055106

1/2:

1.188205267996793599326450395464404902055295091048869481623500537507138719563548660433135758814106242

1/2:

1/4:

0.9939936206449126628631962801821414813654058901350435299671274382998349997779592937134254190052317636

1/2

1/4:

0.9984853272696200099743848566112762083910691110366575078465449092931500792141049429357529107985018759

1/4:

0.9767674412966416556992898109341287401229209907934897294840396409437221212118332156845467292460625117

1/2:

-0.9462397597698016326301759892761538795337435324460417876958263019513586850193138589675060576890870970

1/4:

0.8219142851414615028586218629540107887494521263179519192029354551101370812899149375488582217427546052

1/2

1/4:

0.7687800976725338056202507874643273996701886148757846725474624612391999315720233441586664992680127436

1/2

1/2:

-0.9708072326378321309192277719440383640119939205302374758663061579037779799458462663878238813164968133

1/4:

0.9070451351397571214377438362837486065346256307496255257627449221845993457117497291321456980810698503

1/2:

-0.9114929736758266370187248165187973241328907130927291431758229246230698755972739834288477902606029335

1/2:

1/4:

0.1326042995245017627653668269010481017692358848429800180314896016540607358054313044109646749785786222

1/2

1/4:

0.06657469134440202544444114053371401153844524911893895271397072779580882760444350435062541576478244531

1/4:

0.2610387333723058745117534726391330867553832866339034907981183028651649359661001356696845350211826392

1/2:

0.9796430728792083711989559625268215536033136521341894628530725758198548128876665930436491442109323503

1/4:

-0.6292795486991482302800689031734255572417287146913077574792020642898209096994019250631578556897730596

1/2

1/4:

-0.6692112516442997740559642538653818122677335808215132755021272968603694840977160868022174753894441113

1/2

1/2:

0.9897084001452577406722430403699906728681563049831918110995624223778164167215182797686675739649048171

1/4:

-0.5439436720339353928126744705170150430559302623292861643777156335335498943746996974415419263819546419

1/2:

0.9600875465811428260029810357619344176593891719179950790946187901793150951241609345420622695497775651

1/2:

1/4:

-0.9959948086900586332204265089700652863777950012485689582824915214231499156484772377850671423428506527

1/2

1/4:

-0.9989981206674913385355241360404488792550932539816557718647784181359933446674384857596572209434553727

1/4:

-0.9840167775023841864368983975377991508026756966031847881368629011623046674340284946407480960480255925

Definition

This table gives the DLMF-normalised odd Mathieu function $\mathrm{se}_n(z,q)$ [1]. It is the odd $2\pi$-periodic solution of $w''+(b_n(q)-2q\cos 2z)w=0$ with DLMF's sign and scale, where $b_n(q)$ is the characteristic value paired with $\mathrm{se}_n(z,q)$ in DLMF's indexing.

Parameters

$n$

— order ($n\geq1$)

$q$

— Mathieu parameter ($q\in\mathbb{Q}_{>0}$)

$z/\pi$

— argument divided by $\pi$ ($0\leq z/\pi\leq 1/2$)

Formulas

(1)

The function satisfies $\frac{d^2}{dz^2}\mathrm{se}_n(z,q)+(b_n(q)-2q\cos 2z)\mathrm{se}_n(z,q)=0$ [1].

(2)

$\int_0^{2\pi}\mathrm{se}_n(x,q)^2\,dx=\pi$ [1].

(3)

$\mathrm{se}_n(z,0)=\sin(nz)$ [1].

(4)

$\mathrm{se}_n(-z,q)=-\mathrm{se}_n(z,q)$, and $\mathrm{se}_n(z+\pi,q)=(-1)^n\mathrm{se}_n(z,q)$ [1].

Comments

(5)

The parameter $z$ is measured in radians. The stored parameter value is $z/\pi$, so the row $z/\pi=1/6$ means $z=\pi/6$.

(6)

The sign and scale are fixed by the DLMF normalisation in (2) and by the limiting value $\mathrm{se}_n(z,0)=\sin(nz)$ in (3). This is the same normalisation used by SciPy after converting this table's radian argument to the degrees expected by scipy.special.mathieu_sem [3].

(7)

Values are listed only for $0\leq z\leq\pi/2$. The parity and periodicity formulas in (4) determine the values at the corresponding points in a full period. At $z=0$ all entries are $0$ by oddness. At $z=\pi/2$, the even-order entries are $0$ because they are odd and $\pi$-periodic.

(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("T438")
generator.value({"n": "1", "q": "1", "z": "1/4"}, 100)

Links

[1]

DLMF 28.2: Definitions and Basic Properties

[2]

DLMF 28.4: Fourier Series

[3]

SciPy: scipy.special.mathieu_sem

[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

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Data properties

Entries are of type: real number

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

Table is complete: no (it holds $1\leq n\leq6$ 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.

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.