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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 ${\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1}$, as real functions they parametrize the cubic Fermat curve ${\displaystyle x^{3}+y^{3}=1}$, just as the trigonometric functions sine and cosine parametrize the unit circle ${\displaystyle x^{2}+y^{2}=1}$.

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.

Definition

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

${\displaystyle {\frac {d}{dz}}\operatorname {cm} z=-\operatorname {sm} ^{2}z,\ {\frac {d}{dz}}\operatorname {sm} z=\operatorname {cm} ^{2}z,\ \operatorname {cm} (0)=1,\ \operatorname {sm} (0)=0}$

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

${\displaystyle z=\int _{0}^{\operatorname {sm} z}{\frac {dw}{(1-w^{3})^{2/3}}}=\int _{\operatorname {cm} z}^{1}{\frac {dw}{(1-w^{3})^{2/3}}}}$

which can also be expressed using the hypergeometric function:

${\displaystyle \operatorname {sm} ^{-1}(z)=z\;{}_{2}F_{1}{\bigl (}{\tfrac {1}{3}},{\tfrac {2}{3}};{\tfrac {4}{3}};z^{3}{\bigr )}}$

Parametrization of the cubic Fermat curve

Both sm and cm have a period along the real axis of ${\displaystyle \pi _{3}=\mathrm {B} {\bigl (}{\tfrac {1}{3}},{\tfrac {1}{3}}{\bigr )}={\tfrac {\sqrt {3}}{2\pi }}\Gamma ^{3}{\bigl (}{\tfrac {1}{3}}{\bigr )}\approx 5.29991625}$ with ${\displaystyle \mathrm {B} }$ the beta function and ${\displaystyle \Gamma }$ the gamma function:

${\displaystyle {\begin{aligned}{\tfrac {1}{3}}\pi _{3}&=\int _{-\infty }^{0}{\frac {dx}{(1-x^{3})^{2/3}}}=\int _{0}^{1}{\frac {dx}{(1-x^{3})^{2/3}}}=\int _{1}^{\infty }{\frac {dx}{(1-x^{3})^{2/3}}}\\&\approx 1.76663875\end{aligned}}}$

They satisfy the identity ${\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1}$. The parametric function ${\displaystyle t\mapsto (\operatorname {cm} t,\,\operatorname {sm} t),}$ ${\displaystyle t\in {\bigl }}$ parametrizes the cubic Fermat curve ${\displaystyle x^{3}+y^{3}=1,}$ with ${\displaystyle {\tfrac {1}{2}}t}$ representing the signed area lying between the segment from the origin to ${\displaystyle (1,\,0)}$, the segment from the origin to ${\displaystyle (\operatorname {cm} t,\,\operatorname {sm} t)}$, 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. To see why, apply Green's theorem:

${\displaystyle A={\tfrac {1}{2}}\int _{0}^{t}(x\mathop {dy} -y\mathop {dx} )={\tfrac {1}{2}}\int _{0}^{t}(\operatorname {cm} ^{3}t+\operatorname {sm} ^{3}t)\mathop {dt} ={\tfrac {1}{2}}\int _{0}^{t}dt={\tfrac {1}{2}}t.}$

Notice that the area between the ${\displaystyle x+y=0}$ and ${\displaystyle x^{3}+y^{3}=1}$ can be broken into three pieces, each of area ${\displaystyle {\tfrac {1}{6}}\pi _{3}}$:

${\displaystyle {\begin{aligned}{\tfrac {1}{2}}\pi _{3}&=\int _{-\infty }^{\infty }{\bigl (}(1-x^{3})^{1/3}+x{\bigr )}\mathop {dx} \\{\tfrac {1}{6}}\pi _{3}&=\int _{-\infty }^{0}{\bigl (}(1-x^{3})^{1/3}+x{\bigr )}\mathop {dx} =\int _{0}^{1}(1-x^{3})^{1/3}\mathop {dx} .\end{aligned}}}$

Symmetries

The function ${\displaystyle \operatorname {sm} z}$ has zeros at the complex-valued points ${\displaystyle z={\tfrac {1}{\sqrt {3}}}\pi _{3}i(a+b\omega )}$ for any integers ${\displaystyle a}$ and ${\displaystyle b}$, where ${\displaystyle \omega }$ is a cube root of unity, ${\displaystyle \omega =\exp {\tfrac {2}{3}}i\pi =-{\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i}$ (that is, ${\displaystyle a+b\omega }$ is an Eisenstein integer). The function ${\displaystyle \operatorname {cm} z}$ has zeros at the complex-valued points ${\displaystyle z={\tfrac {1}{3}}\pi _{3}+{\tfrac {1}{\sqrt {3}}}\pi _{3}i(a+b\omega )}$. Both functions have poles at the complex-valued points ${\displaystyle z=-{\tfrac {1}{3}}\pi _{3}+{\tfrac {1}{\sqrt {3}}}\pi _{3}i(a+b\omega )}$.

On the real line, ${\displaystyle \operatorname {sm} x=0\leftrightarrow x\in \pi _{3}\mathbb {Z} }$, which is analogous to ${\displaystyle \sin x=0\leftrightarrow x\in \pi \mathbb {Z} }$.

Fundamental reflections, rotations, and translations

Both cm and sm commute with complex conjugation,

${\displaystyle {\begin{aligned}\operatorname {cm} {\bar {z}}&={\overline {\operatorname {cm} z}},\\\operatorname {sm} {\bar {z}}&={\overline {\operatorname {sm} z}}.\end{aligned}}}$

Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), the Dixon function cm is invariant under ${\textstyle {\tfrac {1}{3}}}$ turn rotations of the complex plane, and ${\textstyle {\tfrac {1}{3}}}$ turn rotations of the domain of sm cause ${\displaystyle {\tfrac {1}{3}}}$ turn rotations of the codomain:

${\displaystyle {\begin{aligned}\operatorname {cm} \omega z&=\operatorname {cm} z=\operatorname {cm} \omega ^{2}z,\\\operatorname {sm} \omega z&=\omega \operatorname {sm} z=\omega ^{2}\operatorname {sm} \omega ^{2}z.\end{aligned}}}$

Each Dixon elliptic function is invariant under translations by the Eisenstein integers ${\displaystyle a+b\omega }$ scaled by ${\displaystyle \pi _{3},}$

${\displaystyle {\begin{aligned}\operatorname {cm} {\bigl (}z+\pi _{3}(a+b\omega ){\bigr )}=\operatorname {cm} z,\\\operatorname {sm} {\bigl (}z+\pi _{3}(a+b\omega ){\bigr )}=\operatorname {sm} z.\end{aligned}}}$

Negation of each of cm and sm is equivalent to a ${\displaystyle {\tfrac {1}{3}}\pi _{3}}$ translation of the other,

${\displaystyle {\begin{aligned}\operatorname {cm} (-z)&={\frac {1}{\operatorname {cm} z}}=\operatorname {sm} {\bigl (}z+{\tfrac {1}{3}}\pi _{3}{\bigr )},\\\operatorname {sm} (-z)&=-{\frac {\operatorname {sm} z}{\operatorname {cm} z}}={\frac {1}{\operatorname {sm} {\bigl (}z-{\tfrac {1}{3}}\pi _{3}{\bigr )}}}=\operatorname {cm} {\bigl (}z+{\tfrac {1}{3}}\pi _{3}{\bigr )}.\end{aligned}}}$

For ${\displaystyle n\in \mathbb {\{} 0,1,2\},}$ translations by ${\displaystyle {\tfrac {1}{3}}\pi _{3}\omega }$ give

${\displaystyle {\begin{aligned}\operatorname {cm} {\bigl (}z+{\tfrac {1}{3}}\omega ^{n}\pi _{3}{\bigr )}&=\omega ^{2n}{\frac {-\operatorname {sm} z}{\operatorname {cm} z}},\\\operatorname {sm} {\bigl (}z+{\tfrac {1}{3}}\omega ^{n}\pi _{3}{\bigr )}&=\omega ^{n}{\frac {1}{\operatorname {cm} z}}.\end{aligned}}}$

Specific values

${\displaystyle z}$	${\displaystyle \operatorname {cm} z}$	${\displaystyle \operatorname {sm} z}$
${\displaystyle {-{\tfrac {1}{3}}}\pi _{3}}$	${\displaystyle \infty }$	${\displaystyle \infty }$
${\displaystyle {-{\tfrac {1}{6}}}\pi _{3}}$	${\displaystyle {\sqrt{2}}}$	${\displaystyle -1}$
${\displaystyle 0}$	${\displaystyle 1}$	${\displaystyle 0}$
${\displaystyle {\tfrac {1}{6}}\pi _{3}}$	${\displaystyle 1{\big /}{\sqrt{2}}}$	${\displaystyle 1{\big /}{\sqrt{2}}}$
${\displaystyle {\tfrac {1}{3}}\pi _{3}}$	${\displaystyle 0}$	${\displaystyle 1}$
${\displaystyle {\tfrac {1}{2}}\pi _{3}}$	${\displaystyle -1}$	${\displaystyle {\sqrt{2}}}$
${\displaystyle {\tfrac {2}{3}}\pi _{3}}$	${\displaystyle \infty }$	${\displaystyle \infty }$

More specific values

${\displaystyle z}$	${\displaystyle \operatorname {cm} z}$	${\displaystyle \operatorname {sm} z}$
${\displaystyle {-{\tfrac {1}{4}}}\pi _{3}}$	${\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}}$	${\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt{4}}}}$
${\displaystyle -{\tfrac {2}{9}}\pi _{3}}$	${\displaystyle {\frac {\sqrt{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}}$	${\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt{3}}}}$
${\displaystyle -{\tfrac {1}{9}}\pi _{3}}$	${\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt{3}}}}$	${\displaystyle -{\frac {\sqrt{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}}$
${\displaystyle -{\tfrac {1}{12}}\pi _{3}}$	${\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt{2}}}}}$	${\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt{2}}}}}$
${\displaystyle {\tfrac {1}{12}}\pi _{3}}$	${\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt{4}}}}$	${\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}}$
${\displaystyle {\tfrac {1}{9}}\pi _{3}}$	${\displaystyle {\frac {\sqrt{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}}$	${\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt{3}}}}$
${\displaystyle {\tfrac {2}{9}}\pi _{3}}$	${\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt{3}}}}$	${\displaystyle {\frac {\sqrt{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}}$
${\displaystyle {\tfrac {1}{4}}\pi _{3}}$	${\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}}$	${\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt{4}}}}$
${\displaystyle {\tfrac {5}{12}}\pi _{3}}$	${\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt{2}}}}}$	${\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt{2}}}}}$
${\displaystyle {\tfrac {4}{9}}\pi _{3}}$	${\displaystyle -{\frac {\sqrt{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}}$	${\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt{3}}}}$
${\displaystyle {\tfrac {5}{9}}\pi _{3}}$	${\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt{3}}}}$	${\displaystyle {\frac {\sqrt{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}}$
${\displaystyle {\tfrac {7}{12}}\pi _{3}}$	${\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt{4}}}}$	${\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}}$

Sum and difference identities

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

${\displaystyle {\begin{aligned}\operatorname {cm} (u+v)&={\frac {\operatorname {sm} u\,\operatorname {cm} u-\operatorname {sm} v\,\operatorname {cm} v}{\operatorname {sm} u\,\operatorname {cm} ^{2}v-\operatorname {cm} ^{2}u\,\operatorname {sm} v}}\\\operatorname {cm} (u-v)&={\frac {\operatorname {cm} ^{2}u\,\operatorname {cm} v-\operatorname {sm} u\,\operatorname {sm} ^{2}v}{\operatorname {cm} u\,\operatorname {cm} ^{2}v-\operatorname {sm} ^{2}u\,\operatorname {sm} v}}\\\operatorname {sm} (u+v)&={\frac {\operatorname {sm} ^{2}u\,\operatorname {cm} v-\operatorname {cm} u\,\operatorname {sm} ^{2}v}{\operatorname {sm} u\,\operatorname {cm} ^{2}v-\operatorname {cm} ^{2}u\,\operatorname {sm} v}}\\\operatorname {sm} (u-v)&={\frac {\operatorname {sm} u\,\operatorname {cm} u-\operatorname {sm} v\,\operatorname {cm} v}{\operatorname {cm} u\,\operatorname {cm} ^{2}v-\operatorname {sm} ^{2}u\,\operatorname {sm} v}}\end{aligned}}}$

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

${\displaystyle {\begin{aligned}\operatorname {cm} (x+\omega y)&={\frac {\operatorname {sm} x\,\operatorname {cm} x-\omega \,\operatorname {sm} y\,\operatorname {cm} y}{\operatorname {sm} x\,\operatorname {cm} ^{2}y-\omega \,\operatorname {cm} ^{2}x\,\operatorname {sm} y}}\\&={\frac {\operatorname {cm} x(\operatorname {sm} ^{2}x\,\operatorname {cm} ^{2}y+\operatorname {cm} x\,\operatorname {sm} ^{2}y\,\operatorname {cm} y+\operatorname {sm} x\,\operatorname {cm} ^{2}x\,\operatorname {sm} y)}{\operatorname {sm} ^{2}x\,\operatorname {cm} ^{4}y+\operatorname {sm} x\,\operatorname {cm} ^{2}x\,\operatorname {sm} y\,\operatorname {cm} ^{2}y+\operatorname {cm} ^{4}x\,\operatorname {sm} ^{2}y}}\\&\qquad +\omega {\frac {\operatorname {sm} x\,\operatorname {sm} y(\operatorname {cm} ^{3}x-\operatorname {cm} ^{3}y)}{\operatorname {sm} ^{2}x\,\operatorname {cm} ^{4}y+\operatorname {sm} x\,\operatorname {cm} ^{2}x\,\operatorname {sm} y\,\operatorname {cm} ^{2}y+\operatorname {cm} ^{4}x\,\operatorname {sm} ^{2}y}}\\\operatorname {sm} (x+\omega y)&={\frac {\operatorname {sm} ^{2}x\,\operatorname {cm} y-\omega ^{2}\,\operatorname {cm} x\,\operatorname {sm} ^{2}y}{\operatorname {sm} x\,\operatorname {cm} ^{2}y-\omega \,\operatorname {cm} ^{2}x\,\operatorname {sm} y}}\\&={\frac {\operatorname {sm} x(\operatorname {sm} x\,\operatorname {cm} x\,\operatorname {cm} ^{2}y+\operatorname {sm} y\,\operatorname {cm} ^{3}x+\operatorname {sm} y\,\operatorname {cm} ^{3}y)}{\operatorname {sm} ^{2}x\,\operatorname {cm} ^{4}y+\operatorname {sm} x\,\operatorname {cm} ^{2}x\,\operatorname {sm} y\,\operatorname {cm} ^{2}y+\operatorname {cm} ^{4}x\,\operatorname {sm} ^{2}y}}\\&\qquad +\omega {\frac {\operatorname {sm} y(\operatorname {sm} x\,\operatorname {cm} ^{3}x+\operatorname {sm} x\,\operatorname {cm} ^{3}y+\operatorname {cm} ^{2}x\,\operatorname {sm} y\,\operatorname {cm} y)}{\operatorname {sm} ^{2}x\,\operatorname {cm} ^{4}y+\operatorname {sm} x\,\operatorname {cm} ^{2}x\,\operatorname {sm} y\,\operatorname {cm} ^{2}y+\operatorname {cm} ^{4}x\,\operatorname {sm} ^{2}y}}\end{aligned}}}$

Multiple-argument identities

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

${\displaystyle {\begin{aligned}\operatorname {cm} 2u&={\frac {\operatorname {cm} ^{3}u-\operatorname {sm} ^{3}u}{\operatorname {cm} u(1+\operatorname {sm} ^{3}u)}}={\frac {2\operatorname {cm} ^{3}u-1}{2\operatorname {cm} u-\operatorname {cm} ^{4}u}},\\\operatorname {sm} 2u&={\frac {\operatorname {sm} u(1+\operatorname {cm} ^{3}u)}{\operatorname {cm} u(1+\operatorname {sm} ^{3}u)}}={\frac {2\operatorname {sm} u-\operatorname {sm} ^{4}u}{2\operatorname {cm} u-\operatorname {cm} ^{4}u}},\\\operatorname {cm} 3u&={\frac {\operatorname {cm} ^{9}u-6\operatorname {cm} ^{6}u+3\operatorname {cm} ^{3}u+1}{\operatorname {cm} ^{9}u+3\operatorname {cm} ^{6}u-6\operatorname {cm} ^{3}u+1}},\\\operatorname {sm} 3u&={\frac {3\operatorname {sm} u\,\operatorname {cm} u(\operatorname {sm} ^{3}u\,\operatorname {cm} ^{3}u-1)}{\operatorname {cm} ^{9}u+3\operatorname {cm} ^{6}u-6\operatorname {cm} ^{3}u+1}}.\end{aligned}}}$

From these formulas it can be deduced that expressions in form ${\displaystyle \operatorname {cm} ({\frac {k\pi _{3}}{2^{n}3^{m}}})}$ and ${\displaystyle \operatorname {sm} ({\frac {k\pi _{3}}{2^{n}3^{m}}})}$ are either signless infinities, or origami-constructibles for any ${\displaystyle n,m,k\in \mathbb {N} }$ (In this paragraph, ${\displaystyle \mathbb {M} =}$ set of all origami-constructibles ${\displaystyle \cup \{\infty }\}$). Because by finding ${\displaystyle \operatorname {cm} ({\frac {x}{2}})}$, quartic or lesser degree in some cases equation has to be solved as seen from duplication formula which means that if ${\displaystyle \operatorname {cm} x\in \mathbb {M} }$, then ${\displaystyle \operatorname {cm} ({\frac {x}{2}})\in \mathbb {M} }$. To find one-third of argument value of cm, equation which is reductible to cubic or lesser degree in some cases by variable exchange ${\displaystyle t=x^{3}}$ has to be solved as seen from triplication formula from that follows: if ${\displaystyle \operatorname {cm} x\in \mathbb {M} }$ then ${\displaystyle \operatorname {cm} ({\frac {x}{3}})\in \mathbb {M} }$ is true. Statement ${\displaystyle \operatorname {cm} x\in \mathbb {M} }$ ${\displaystyle \Rightarrow }$ ${\displaystyle \operatorname {cm} (nx)\in \mathbb {M} }$ is true, because any multiple argument formula is a rational function. If ${\displaystyle \operatorname {cm} x\in \mathbb {M} }$, then ${\displaystyle \operatorname {sm} x\in \mathbb {M} }$ because ${\displaystyle \operatorname {sm} x=\omega ^{p}\,{\sqrt{1-\operatorname {cm} ^{3}x}}}$ where ${\displaystyle p\in \{0,1,2\}}$.

Specific value identities

The ${\displaystyle \operatorname {cm} }$ function satisfies the identities $${\displaystyle {\begin{aligned}\operatorname {cm} {\tfrac {2}{9}}\pi _{3}&=-\operatorname {cm} {\tfrac {1}{9}}\pi _{3}\,\operatorname {cm} {\tfrac {4}{9}}\pi _{3},\\\operatorname {cm} {\tfrac {1}{4}}\pi _{3}&=\operatorname {cl} {\tfrac {1}{3}}\varpi ,\end{aligned}}}$$

where ${\displaystyle \operatorname {cl} }$ is lemniscate cosine and ${\displaystyle \varpi }$ is Lemniscate constant.

Power series

The cm and sm functions can be approximated for ${\displaystyle |z|<{\tfrac {1}{3}}\pi _{3}}$ by the Taylor series

${\displaystyle {\begin{aligned}\operatorname {cm} z&=c_{0}+c_{1}z^{3}+c_{2}z^{6}+c_{3}z^{9}+\cdots +c_{n}z^{3n}+\cdots \\\operatorname {sm} z&=s_{0}z+s_{1}z^{4}+s_{2}z^{7}+s_{3}z^{10}+\cdots +s_{n}z^{3n+1}+\cdots \end{aligned}}}$

whose coefficients satisfy the recurrence ${\displaystyle c_{0}=s_{0}=1,}$

${\displaystyle {\begin{aligned}c_{n}&=-{\frac {1}{3n}}\sum _{k=0}^{n-1}s_{k}s_{n-1-k}\\s_{n}&={\frac {1}{3n+1}}\sum _{k=0}^{n}c_{k}c_{n-k}\end{aligned}}}$

These recurrences result in:

${\displaystyle {\begin{aligned}\operatorname {cm} z&=1-{\frac {1}{3}}z^{3}+{\frac {1}{18}}z^{6}-{\frac {23}{2268}}z^{9}+{\frac {25}{13608}}z^{12}-{\frac {619}{1857492}}z^{15}+\cdots \\\operatorname {sm} z&=z-{\frac {1}{6}}z^{4}+{\frac {2}{63}}z^{7}-{\frac {13}{2268}}z^{10}+{\frac {23}{22113}}z^{13}-{\frac {2803}{14859936}}z^{16}+\cdots \end{aligned}}}$

Relation to other elliptic functions

Weierstrass elliptic function

The equianharmonic Weierstrass elliptic function ${\displaystyle \wp (z)=\wp {\bigl (}z;0,{\tfrac {1}{27}}{\bigr )},}$ with lattice ${\displaystyle \Lambda =\pi _{3}\mathbb {Z} \oplus \pi _{3}\omega \mathbb {Z} }$ a scaling of the Eisenstein integers, can be defined as:

${\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \setminus \{0\}}\!\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)}$

The function ${\displaystyle \wp (z)}$ solves the differential equation:

${\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-{\tfrac {1}{27}}}$

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

${\displaystyle z=\int _{\infty }^{\wp (z)}{\frac {dw}{\sqrt {4w^{3}-{\tfrac {1}{27}}}}}}$

In terms of ${\displaystyle \wp (z)}$, the Dixon elliptic functions can be written:

${\displaystyle \operatorname {cm} z={\frac {3\wp '(z)+1}{3\wp '(z)-1}},\ \operatorname {sm} z={\frac {-6\wp (z)}{3\wp '(z)-1}}}$

Likewise, the Weierstrass elliptic function ${\displaystyle \wp (z)=\wp {\bigl (}z;0,{\tfrac {1}{27}}{\bigr )}}$ can be written in terms of Dixon elliptic functions:

${\displaystyle \wp '(z)={\frac {\operatorname {cm} z+1}{3(\operatorname {cm} z-1)}},\ \wp (z)={\frac {-\operatorname {sm} z}{3(\operatorname {cm} z-1)}}}$

Jacobi elliptic functions

The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley. Let ${\displaystyle k=e^{5i\pi /6}}$, ${\displaystyle \theta =3^{\frac {1}{4}}e^{5i\pi /12}}$, ${\displaystyle s=\operatorname {sn} (u,k)}$, ${\displaystyle c=\operatorname {cn} (u,k)}$, and ${\displaystyle d=\operatorname {dn} (u,k)}$. Then, let

${\displaystyle \xi (u)={\frac {-1+\theta scd}{1+\theta scd}}}$, ${\displaystyle \eta (u)={\frac {2^{1/3}\left(1+\theta ^{2}s^{2}\right)}{1+\theta scd}}}$.

Finally, the Dixon elliptic functions are as so:

${\displaystyle \operatorname {sm} (z)=\xi \left({\frac {z+\pi _{3}/6}{2^{1/3}\theta }}\right)}$, ${\displaystyle \operatorname {cm} (z)=\eta \left({\frac {z+\pi _{3}/6}{2^{1/3}\theta }}\right)}$.

Generalized trigonometry

Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an ${\displaystyle n=2}$ case, and the functions sm and cm as an ${\displaystyle n=3}$ case.

For example, defining ${\displaystyle \pi _{n}=\mathrm {B} {\bigl (}{\tfrac {1}{n}},{\tfrac {1}{n}}{\bigr )}}$ and ${\displaystyle \sin _{n}z,\,\cos _{n}z}$ the inverses of an integral:

${\displaystyle z=\int _{0}^{\sin _{n}z}{\frac {dw}{(1-w^{n})^{(n-1)/n}}}=\int _{\cos _{n}z}^{1}{\frac {dw}{(1-w^{n})^{(n-1)/n}}}}$

The area in the positive quadrant under the curve ${\displaystyle x^{n}+y^{n}=1}$ is

${\displaystyle \int _{0}^{1}(1-x^{n})^{1/n}\mathop {dx} ={\frac {\pi _{n}}{2n}}}$.

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

Applications

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.

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 ${\displaystyle W=\operatorname {sm} ,\ W_{1}=\operatorname {cm} .}$
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 ${\displaystyle 3w}$. Dixon (1890) calls it ${\displaystyle 3\lambda }$; Adams (1925) and Robinson (2019) each call it ${\displaystyle 3K}$. Van Fossen Conrad & Flajolet (2005) call it ${\displaystyle \pi _{3}}$. 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 ${\displaystyle \operatorname {sm} z}$ goes from ${\displaystyle -\pi }$ to ${\displaystyle \pi }$, the colors go through cyan, blue (${\displaystyle \operatorname {Arg} \approx -\pi /2}$), magneta, red (${\displaystyle \operatorname {Arg} \approx 0}$), orange, yellow (${\displaystyle \operatorname {Arg} \approx \pi /2}$), green, and back to cyan (${\displaystyle \operatorname {Arg} \approx \pi }$).
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 ${\displaystyle \wp (z;0,-1).}$
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 ${\textstyle \int dx/(1-x^{3}){}^{2/3}}$ 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 x + y − 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 x + y − 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 Cartographer 13.
• 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
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• 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 
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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 ${\displaystyle x^{3}+y^{3}=z^{3}}$, 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/

• Complex analysis
• Elliptic functions