Dixon elliptic functions

The Dixon elliptic functions cm, sm applied to a real-valued argument x. Both functions are periodic with real period π3 ≈ 5.29991625

In mathematics, the Dixon elliptic functions sm and cm are two elliptic functions (doubly periodic meromorphic functions on the complex plane) that map from each regular hexagon in a hexagonal tiling to the whole complex plane. Because these functions satisfy the identity cm3z+sm3z=1, as real functions they parametrize the cubic Fermat curve x3+y3=1, just as the trigonometric functions sine and cosine parametrize the unit circle x2+y2=1.

They were named sm and cm by Alfred Dixon in 1890, by analogy to the trigonometric functions sine and cosine and the Jacobi elliptic functions sn and cn; Göran Dillner described them earlier in 1873.[1]

## Contents

* 1 Definition
* 2 Parametrization of the cubic Fermat curve
* 3 Symmetries
* 3.1 Fundamental reflections, rotations, and translations
* 3.2 Specific values
* 3.2.1 More specific values
* 4 Sum and difference identities
* 5 Multiple-argument identities
* 6 Specific value identities
* 7 Power series
* 8 Relation to other elliptic functions
* 8.1 Weierstrass elliptic function
* 8.2 Jacobi elliptic functions
* 9 Generalized trigonometry
* 10 Applications
* 11 See also
* 12 Notes
* 13 References
* 14 External links

## Definition

The functions sm and cm can be defined as the solutions to the initial value problem:[2]

ddzcm⁡z=−sm2z, ddzsm⁡z=cm2z, cm⁡(0)=1, sm⁡(0)=0

Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:[3]

z=∫0sm⁡zdw(1−w3)2/3=∫cm⁡z1dw(1−w3)2/3

which can also be expressed using the hypergeometric function:[4]

sm−1(z)=z2F1(13,23;43;z3)

## Parametrization of the cubic Fermat curve

The function t ↦ (cm t, sm t) parametrizes the cubic Fermat curve, with area of the sector equal to half the argument t.

Both sm and cm have a period along the real axis of π3=B(13,13)=32πΓ3(13)≈5.29991625 with B the beta function and Γ the gamma function:[5]

13π3=∫−∞0dx(1−x3)2/3=∫01dx(1−x3)2/3=∫1∞dx(1−x3)2/3≈1.76663875

They satisfy the identity cm3z+sm3z=1. The parametric function t↦(cm⁡t,sm⁡t), t∈[−13π3,23π3] parametrizes the cubic Fermat curve x3+y3=1, with 12t representing the signed area lying between the segment from the origin to (1,0), the segment from the origin to (cm⁡t,sm⁡t), and the Fermat curve, analogous to the relationship between the argument of the trigonometric functions and the area of a sector of the unit circle.[6] To see why, apply Green's theorem:

A=12∫0t(xdy−ydx)=12∫0t(cm3t+sm3t)dt=12∫0tdt=12t.

Notice that the area between the x+y=0 and x3+y3=1 can be broken into three pieces, each of area 16π3:

12π3=∫−∞∞((1−x3)1/3+x)dx16π3=∫−∞0((1−x3)1/3+x)dx=∫01(1−x3)1/3dx.

## Symmetries

The Dixon elliptic function sm z in the complex plane, illustrating its double periodicity (ω = e2πi/3).[7]

The function sm⁡z has zeros at the complex-valued points z=13π3i(a+bω) for any integers a and b, where ω is a cube root of unity, ω=exp⁡23iπ=−12+32i (that is, a+bω is an Eisenstein integer). The function cm⁡z has zeros at the complex-valued points z=13π3+13π3i(a+bω). Both functions have poles at the complex-valued points z=−13π3+13π3i(a+bω).

On the real line, sm⁡x=0↔x∈π3ℤ, which is analogous to sin⁡x=0↔x∈πℤ.

### Fundamental reflections, rotations, and translations

Both cm and sm commute with complex conjugation,

cm⁡z¯=cm⁡z‾,sm⁡z¯=sm⁡z‾.

Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), the Dixon function cm is invariant under 13 turn rotations of the complex plane, and 13 turn rotations of the domain of sm cause 13 turn rotations of the codomain:

cm⁡ωz=cm⁡z=cm⁡ω2z,sm⁡ωz=ωsm⁡z=ω2sm⁡ω2z.

Each Dixon elliptic function is invariant under translations by the Eisenstein integers a+bω scaled by π3,

cm⁡(z+π3(a+bω))=cm⁡z,sm⁡(z+π3(a+bω))=sm⁡z.

Negation of each of cm and sm is equivalent to a 13π3 translation of the other,

cm⁡(−z)=1cm⁡z=sm⁡(z+13π3),sm⁡(−z)=−sm⁡zcm⁡z=1sm⁡(z−13π3)=cm⁡(z+13π3).

For n∈{0,1,2}, translations by 13π3ω give

cm⁡(z+13ωnπ3)=ω2n−sm⁡zcm⁡z,sm⁡(z+13ωnπ3)=ωn1cm⁡z.

### Specific values

z | cm⁡z | sm⁡z

|
|

−13π3 | ∞ | ∞
−16π3 | 23 | −1
0 | 1 | 0
16π3 | 1/23 | 1/23
13π3 | 0 | 1
12π3 | −1 | 23
23π3 | ∞ | ∞

#### More specific values

z | cm⁡z | sm⁡z

|
|

−14π3 | 1+3+232 | −1−3+2343
−112π3 | −1+3+23223 | −1+3−23223
112π3 | −1+3+2343 | 1+3−232
14π3 | 1+3−232 | −1+3+2343
512π3 | −1+3−23223 | −1+3+23223
712π3 | −1−3+2343 | 1+3+232

## Sum and difference identities

The Dixon elliptic functions satisfy the argument sum and difference identities:[8]

cm⁡(u+v)=sm⁡ucm⁡u−sm⁡vcm⁡vsm⁡ucm2v−cm2usm⁡vcm⁡(u−v)=cm2ucm⁡v−sm⁡usm2vcm⁡ucm2v−sm2usm⁡vsm⁡(u+v)=sm2ucm⁡v−cm⁡usm2vsm⁡ucm2v−cm2usm⁡vsm⁡(u−v)=sm⁡ucm⁡u−sm⁡vcm⁡vcm⁡ucm2v−sm2usm⁡v

These formulas can be used to compute the complex-valued functions in real components:[citation needed]

cm⁡(x+ωy)=sm⁡xcm⁡x−ωsm⁡ycm⁡ysm⁡xcm2y−ωcm2xsm⁡y=cm⁡x(sm2xcm2y+cm⁡xsm2ycm⁡y+sm⁡xcm2xsm⁡y)sm2xcm4y+sm⁡xcm2xsm⁡ycm2y+cm4xsm2y+ωsm⁡xsm⁡y(cm3x−cm3y)sm2xcm4y+sm⁡xcm2xsm⁡ycm2y+cm4xsm2ysm⁡(x+ωy)=sm2xcm⁡y−ω2cm⁡xsm2ysm⁡xcm2y−ωcm2xsm⁡y=sm⁡x(sm⁡xcm⁡xcm2y+sm⁡ycm3x+sm⁡ycm3y)sm2xcm4y+sm⁡xcm2xsm⁡ycm2y+cm4xsm2y+ωsm⁡y(sm⁡xcm3x+sm⁡xcm3y+cm2xsm⁡ycm⁡y)sm2xcm4y+sm⁡xcm2xsm⁡ycm2y+cm4xsm2y

## Multiple-argument identities

Argument duplication and triplication identities can be derived from the sum identity:[9]

cm⁡2u=cm3u−sm3ucm⁡u(1+sm3u)=2cm3u−12cm⁡u−cm4u,sm⁡2u=sm⁡u(1+cm3u)cm⁡u(1+sm3u)=2sm⁡u−sm4u2cm⁡u−cm4u,cm⁡3u=cm9u−6cm6u+3cm3u+1cm9u+3cm6u−6cm3u+1,sm⁡3u=3sm⁡ucm⁡u(sm3ucm3u−1)cm9u+3cm6u−6cm3u+1.

From these formulas it can be deduced that expressions in form cm⁡(kπ32n3m) and sm⁡(kπ32n3m) are either signless infinities, or origami-constructibles for any n,m,k∈ℕ (In this paragraph, 𝕄= set of all origami-constructibles ∪{∞}). Because by finding cm⁡(x2), quartic or lesser degree in some cases equation has to be solved as seen from duplication formula which means that if cm⁡x∈𝕄, then cm⁡(x2)∈𝕄. To find one-third of argument value of cm, equation which is reductible to cubic or lesser degree in some cases by variable exchange t=x3 has to be solved as seen from triplication formula from that follows: if cm⁡x∈𝕄 then cm⁡(x3)∈𝕄 is true. Statement cm⁡x∈𝕄 ⇒ cm⁡(nx)∈𝕄 is true, because any multiple argument formula is a rational function. If cm⁡x∈𝕄, then sm⁡x∈𝕄 because sm⁡x=ωp1−cm3x3 where p∈{0,1,2}.

## Specific value identities

The cm function satisfies the identities cm⁡29π3=−cm⁡19π3cm⁡49π3,cm⁡14π3=cl⁡13ϖ,

where cl is lemniscate cosine and ϖ is Lemniscate constant.[citation needed]

## Power series

The cm and sm functions can be approximated for |z|<13π3 by the Taylor series

cm⁡z=c0+c1z3+c2z6+c3z9+⋯+cnz3n+⋯sm⁡z=s0z+s1z4+s2z7+s3z10+⋯+snz3n+1+⋯

whose coefficients satisfy the recurrence c0=s0=1,[10]

cn=−13n∑k=0n−1sksn−1−ksn=13n+1∑k=0nckcn−k

These recurrences result in:[11]

cm⁡z=1−13z3+118z6−232268z9+2513608z12−6191857492z15+⋯sm⁡z=z−16z4+263z7−132268z10+2322113z13−280314859936z16+⋯

## Relation to other elliptic functions

### Weierstrass elliptic function

Elliptic curve y2=4x3−127 for the Weierstrass ℘-function z↦℘(z;0,127) related to the Dixon elliptic functions.

The equianharmonic Weierstrass elliptic function ℘(z)=℘(z;0,127), with lattice Λ=π3ℤ⊕π3ωℤ a scaling of the Eisenstein integers, can be defined as:[12]

℘(z)=1z2+∑λ∈Λ∖{0}(1(z−λ)2−1λ2)

The function ℘(z) solves the differential equation:

℘′(z)2=4℘(z)3−127

We can also write it as the inverse of the integral:

z=∫∞℘(z)dw4w3−127

In terms of ℘(z), the Dixon elliptic functions can be written:[13]

cm⁡z=3℘′(z)+13℘′(z)−1, sm⁡z=−6℘(z)3℘′(z)−1

Likewise, the Weierstrass elliptic function ℘(z)=℘(z;0,127) can be written in terms of Dixon elliptic functions:

℘′(z)=cm⁡z+13(cm⁡z−1), ℘(z)=−sm⁡z3(cm⁡z−1)

### Jacobi elliptic functions

The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley.[14] Let k=e5iπ/6, θ=314e5iπ/12, s=sn⁡(u,k), c=cn⁡(u,k), and d=dn⁡(u,k). Then, let

ξ(u)=−1+θscd1+θscd, η(u)=21/3(1+θ2s2)1+θscd.

Finally, the Dixon elliptic functions are as so:

sm⁡(z)=ξ(z+π3/621/3θ), cm⁡(z)=η(z+π3/621/3θ).

## Generalized trigonometry

Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an n=2 case, and the functions sm and cm as an n=3 case.[15]

For example, defining πn=B(1n,1n) and sinnz,cosnz the inverses of an integral:

z=∫0sinnzdw(1−wn)(n−1)/n=∫cosnz1dw(1−wn)(n−1)/n

The area in the positive quadrant under the curve xn+yn=1 is

∫01(1−xn)1/ndx=πn2n.

The quartic n=4 case results in a square lattice in the complex plane, related to the lemniscate elliptic functions.

## Applications

A conformal map projection of the globe onto an octahedron. Because the octahedron has equilateral triangle faces, this projection can be described in terms of sm and cm functions.

The Dixon elliptic functions are conformal maps from an equilateral triangle to a disk, and are therefore helpful for constructing polyhedral conformal map projections involving equilateral triangles, for example projecting the sphere onto a triangle, hexagon, tetrahedron, octahedron, or icosahedron.[16]

## See also

* Eisenstein integer
* Elliptic function
* Abel elliptic functions
* Jacobi elliptic functions
* Lemniscate elliptic functions
* Weierstrass elliptic function
* Lee conformal world in a tetrahedron
* Schwarz–Christoffel mapping

## Notes

1. ↑ Dixon (1890), Dillner (1873). Dillner uses the symbols W=sm⁡, W1=cm⁡.
2. ↑ Dixon (1890), Van Fossen Conrad & Flajolet (2005), Robinson (2019).
3. ↑ The mapping for a general regular polygon is described in Schwarz (1869).
4. ↑ van Fossen Conrad & Flajolet (2005) p. 6.
5. ↑ Dillner (1873) calls the period 3w. Dixon (1890) calls it 3λ; Adams (1925) and Robinson (2019) each call it 3K. Van Fossen Conrad & Flajolet (2005) call it π3. Also see OEIS A197374.
6. ↑ Dixon (1890), Van Fossen Conrad & Flajolet (2005)
7. ↑ Dark areas represent zeros, and bright areas represent poles. As the argument of sm⁡z goes from −π to π, the colors go through cyan, blue (Arg⁡≈−π/2), magneta, red (Arg⁡≈0), orange, yellow (Arg⁡≈π/2), green, and back to cyan (Arg⁡≈π).
8. ↑ Dixon (1890), Adams (1925)
9. ↑ Dixon (1890), p. 185–186. Robinson (2019).
10. ↑ Adams (1925)
11. ↑ van Fossen Conrad & Flajolet (2005). Also see OEIS A104133, A104134.
12. ↑ Reinhardt & Walker (2010)
13. ↑ Chapling (2018), Robinson (2019). Adams (1925) instead expresses the Dixon elliptic functions in terms of the Weierstrass elliptic function ℘(z;0,−1).
14. ↑ van Fossen Conrad & Flajolet (2005), p.38
15. ↑ Lundberg (1879), Grammel (1948), Shelupsky (1959), Burgoyne (1964), Gambini, Nicoletti, & Ritelli (2021).
16. ↑ Adams (1925), Cox (1935), Magis (1938), Lee (1973), Lee (1976), McIlroy (2011), Chapling (2016).

## References

* O. S. Adams (1925). Elliptic functions applied to conformal world maps (No. 297). US Government Printing Office. ftp://ftp.library.noaa.gov/docs.lib/htdocs/rescue/cgs_specpubs/QB275U35no1121925.pdf
* R. Bacher & P. Flajolet (2010) “Pseudo-factorials, elliptic functions, and continued fractions” The Ramanujan journal 21(1), 71–97. https://arxiv.org/pdf/0901.1379.pdf
* A. Cayley (1882) “Reduction of ∫dx/(1−x3)2/3 to elliptic integrals”. Messenger of Mathematics 11, 142–143. https://gdz.sub.uni-goettingen.de/id/PPN599484047_0011?tify={"pages":[146]}
* F. D. Burgoyne (1964) “Generalized trigonometric functions”. Mathematics of Computation 18(86), 314–316. https://www.jstor.org/stable/2003310
* A. Cayley (1883) “On the elliptic function solution of the equation x3 \+ y3 − 1 = 0”, Proceedings of the Cambridge Philosophical Society 4, 106–109. https://archive.org/details/proceedingsofcam4188083camb/page/106/
* R. Chapling (2016) “Invariant Meromorphic Functions on the Wallpaper Groups”. https://arxiv.org/pdf/1608.05677
* J. F. Cox (1935) “Répresentation de la surface entière de la terre dans une triangle équilatéral”, Bulletin de la Classe des Sciences, Académie Royale de Belgique 5e, 21, 66–71.
* G. Dillner (1873) “Traité de calcul géométrique supérieur”, Chapter 16, Nova acta Regiae Societatis Scientiarum Upsaliensis, Ser. III 8, 94–102. https://archive.org/details/novaactaregiaeso38kung/page/94/
* Dixon, A. C. (1890). "On the doubly periodic functions arising out of the curve x3 \+ y3 − 3αxy = 1". Quarterly Journal of Pure and Applied Mathematics XXIV: 167–233. https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify={"pages":[179]}.''x''3+++''y''3+−+3''αxy''+=+1&rft.jtitle=Quarterly+Journal+of+Pure+and+Applied+Mathematics&rft.aulast=Dixon&rft.aufirst=A.+C.&rft.au=Dixon, A.+C.&rft.date=1890&rft.volume=XXIV&rft.pages=167–233&rft_id=https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify={%22pages%22:%5b179%5d}&rfr_id=info:sid/en.wikibooks.org:Dixon_elliptic_functions">
* A. Dixon (1894) The elementary properties of the elliptic functions. MacMillian. https://archive.org/details/elempropellipt00dixorich/
* Van Fossen Conrad, Eric (2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion". Séminaire Lotharingien de Combinatoire 54: Art. B54g, 44. Bibcode: 2005math......7268V.
* A. Gambini, G. Nicoletti, & D. Ritelli (2021) “Keplerian trigonometry”. Monatshefte für Mathematik 195(1), 55–72. https://doi.org/10.1007/s00605-021-01512-0
* R. Grammel (1948) “Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen”. Archiv der Mathematik 1(1), 47–51. https://doi.org/10.1007/BF02038206
* J. C. Langer & D. A. Singer (2014) “The Trefoil”. Milan Journal of Mathematics 82(1), 161–182. https://case.edu/artsci/math/langer/jlpreprints/Trefoil.pdf
* M. Laurent (1949) “Tables de la fonction elliptique de Dixon pour l’intervalle 0-0, 1030”. Bulletin de l’Académie Royale des Sciences de Belgique Classe des Sciences, 35, 439–450.
* L. P. Lee (1973) “The Conformal Tetrahedric Projection with some Practical Applications”. The Cartographic Journal, 10(1), 22–28. https://doi.org/10.1179/caj.1973.10.1.22
* L. P. Lee (1976) Conformal Projections Based on Elliptic Functions. University of Toronto Press. Cartographica Monograph No. 16.
* E. Lundberg (1879) “Om hypergoniometriska funktioner af komplexa variabla”. Manuscript, 1879. Translation by Jaak Peetre “On hypergoniometric functions of complex variables”. https://web.archive.org/web/20161024183030/http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps
* J. Magis (1938) “Calcul du canevas de la représentation conforme de la sphère entière dans un triangle équilatéral”. Bulletin Géodésique 59(1), 247–256. http://doi.org/10.1007/BF03029866
* M. D. McIlroy (2011) “Wallpaper maps”. Dependable and Historic Computing. Springer. 358–375. https://link.springer.com/chapter/10.1007/978-3-642-24541-1_27
* W. P. Reinhardt & P. L. Walker (2010) “Weierstrass Elliptic and Modular Functions”, NIST Digital Library of Mathematical Functions, §23.5(v). https://dlmf.nist.gov/23.5#v
* P. L. Robinson (2019) “The Dixonian elliptic functions”. https://arxiv.org/abs/1901.04296
* H. A. Schwarz (1869) “Ueber einige Abbildungsaufgaben”. Crelles Journal 1869(70), 105–120. http://doi.org/10.1515/crll.1869.70.105
* B. R. Seth & F. P. White (1934) “Torsion of beams whose cross-section is a regular polygon of n sides”. Mathematical Proceedings of the Cambridge Philosophical Society, 30(2), 139. http://doi.org/10.1017/s0305004100016558
* D. Shelupsky (1959) “A generalization of the trigonometric functions”. The American Mathematical Monthly 66(10), 879–884. https://www.jstor.org/stable/2309789

## External links

* Desmos plots:
* Real-valued Dixon elliptic functions https://www.desmos.com/calculator/5s4gdcnxh2.
* Parametrizing the cubic Fermat curve, https://www.desmos.com/calculator/elqqf4nwas
* On-Line Encyclopedia of Integer Sequences pages:
* “Coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the Dixon elliptic function sm(x,0).” https://oeis.org/A104133
* “Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0).” https://oeis.org/A104134
* “Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z).” https://oeis.org/A197374
* Mathematics Stack Exchange discussions:
* “On x3+y3=z3, the Dixonian elliptic functions, and the Borwein cubic theta functions”, https://math.stackexchange.com/q/2090523/
* “doubly periodic functions as tessellations (other than parallelograms)”, https://math.stackexchange.com/q/35671/

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