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 , as real functions they parametrize the cubic Fermat curve , just as the trigonometric functions sine and cosine parametrize the unit circle .
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]
Definition
The functions sm and cm can be defined as the solutions to the initial value problem:[2]
Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:[3]
which can also be expressed using the hypergeometric function:[4]
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 with the beta function and the gamma function:[5]
They satisfy the identity . The parametric function parametrizes the cubic Fermat curve with representing the signed area lying between the segment from the origin to , the segment from the origin to , 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:
Notice that the area between the and can be broken into three pieces, each of area :
Symmetries
The Dixon elliptic function sm z in the complex plane, illustrating its double periodicity (ω = e2πi/3).[7]
The function has zeros at the complex-valued points for any integers and , where is a cube root of unity, (that is, is an Eisenstein integer). The function has zeros at the complex-valued points . Both functions have poles at the complex-valued points .
On the real line, , which is analogous to .
Fundamental reflections, rotations, and translations
Both cm and sm commute with complex conjugation,
Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), the Dixon function cm is invariant under turn rotations of the complex plane, and turn rotations of the domain of sm cause turn rotations of the codomain:
Each Dixon elliptic function is invariant under translations by the Eisenstein integers scaled by
Negation of each of cm and sm is equivalent to a translation of the other,
For translations by give
Specific values
More specific values
Sum and difference identities
The Dixon elliptic functions satisfy the argument sum and difference identities:[8]
These formulas can be used to compute the complex-valued functions in real components:[citation needed]
Multiple-argument identities
Argument duplication and triplication identities can be derived from the sum identity:[9]
From these formulas it can be deduced that expressions in form and are either signless infinities, or origami-constructibles for any (In this paragraph, set of all origami-constructibles ). Because by finding , quartic or lesser degree in some cases equation has to be solved as seen from duplication formula which means that if , then . To find one-third of argument value of cm, equation which is reductible to cubic or lesser degree in some cases by variable exchange has to be solved as seen from triplication formula from that follows: if then is true. Statement is true, because any multiple argument formula is a rational function. If , then because where .
We can also write it as the inverse of the integral:
In terms of , the Dixon elliptic functions can be written:[13]
Likewise, the Weierstrass elliptic function can be written in terms of Dixon elliptic functions:
Jacobi elliptic functions
The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley.[14] Let , , , , and . Then, let
, .
Finally, the Dixon elliptic functions are as so:
, .
Generalized trigonometry
Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an case, and the functions sm and cm as an case.[15]
For example, defining and the inverses of an integral:
The area in the positive quadrant under the curve is
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]
^Dillner (1873) calls the period . Dixon (1890) calls it ; Adams (1925) and Robinson (2019) each call it . Van Fossen Conrad & Flajolet (2005) call it . Also see OEIS A197374.
^Dixon (1890), Van Fossen Conrad & Flajolet (2005)
^Dark areas represent zeros, and bright areas represent poles. As the argument of goes from to , the colors go through cyan, blue (), magneta, red (), orange, yellow (), green, and back to cyan ().
^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 to elliptic integrals”. Messenger of Mathematics 11, 142–143. https://gdz.sub.uni-goettingen.de/id/PPN599484047_0011?tify={%22pages%22:%5b146%5d}
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.
A. Dixon (1894) The elementary properties of the elliptic functions. MacMillian. https://archive.org/details/elempropellipt00dixorich/
Van Fossen Conrad, Eric; Flajolet, Philippe (2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion". Séminaire Lotharingien de Combinatoire. 54: Art. B54g, 44. arXiv:math/0507268. Bibcode:2005math......7268V. MR 2223029.
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. Toronto: B. V. Gutsell, York University. Cartographica Monographs No. 16. ISBN 0-919870-16-3. Supplement No. 1 to The Canadian Cartographer13.
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
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