Dixon elliptic functions

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.^[1]

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

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

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

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

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

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

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:^[5]

${\displaystyle {\begin{aligned}{\tfrac {1}{3}}\pi _{3}&=\int _{-\infty }^{0}{\frac {dx}{(1-x^{3})^{2/3}}}=\int _{0}^{1}{\frac {dx}{(1-x^{3})^{2/3}}}=\int _{1}^{\infty }{\frac {dx}{(1-x^{3})^{2/3}}}\\[8mu]&\approx 1.76663875\end{aligned}}}$

They satisfy the identity ${\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1}$. The parametric function ${\displaystyle t\mapsto (\operatorname {cm} t,\,\operatorname {sm} t),}$ ${\displaystyle t\in {\bigl [}{-{\tfrac {1}{3}}}\pi _{3},{\tfrac {2}{3}}\pi _{3}{\bigr ]}}$ 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.^[6] 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 ${\displaystyle {\tfrac {1}{6}}\pi _{3}}$:

${\displaystyle {\begin{aligned}{\tfrac {1}{2}}\pi _{3}&=\int _{-\infty }^{\infty }{\bigl (}(1-x^{3})^{1/3}+x{\bigr )}\mathop {dx} \\[8mu]{\tfrac {1}{6}}\pi _{3}&=\int _{-\infty }^{0}{\bigl (}(1-x^{3})^{1/3}+x{\bigr )}\mathop {dx} =\int _{0}^{1}(1-x^{3})^{1/3}\mathop {dx} .\end{aligned}}}$

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 ${\displaystyle \sin x=0\leftrightarrow x\in \pi \mathbb {Z} }$.

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

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

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

${\displaystyle z}$	${\displaystyle \operatorname {cm} z}$	${\displaystyle \operatorname {sm} z}$
${\displaystyle {-{\tfrac {1}{4}}}\pi _{3}}$	${\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}}$	${\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}}$
${\displaystyle -{\tfrac {2}{9}}\pi _{3}}$	${\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}}$	${\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt[{6}]{3}}}}$
${\displaystyle -{\tfrac {1}{9}}\pi _{3}}$	${\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt[{6}]{3}}}}$	${\displaystyle -{\frac {\sqrt[{6}]{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}}$
${\displaystyle -{\tfrac {1}{12}}\pi _{3}}$	${\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}}$	${\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}}$
${\displaystyle {\tfrac {1}{12}}\pi _{3}}$	${\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}}$	${\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}}$
${\displaystyle {\tfrac {1}{9}}\pi _{3}}$	${\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}}$	${\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt[{6}]{3}}}}$
${\displaystyle {\tfrac {2}{9}}\pi _{3}}$	${\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt[{6}]{3}}}}$	${\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}}$
${\displaystyle {\tfrac {1}{4}}\pi _{3}}$	${\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}}$	${\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}}$
${\displaystyle {\tfrac {5}{12}}\pi _{3}}$	${\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}}$	${\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}}$
${\displaystyle {\tfrac {4}{9}}\pi _{3}}$	${\displaystyle -{\frac {\sqrt[{6}]{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}}$	${\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt[{6}]{3}}}}$
${\displaystyle {\tfrac {5}{9}}\pi _{3}}$	${\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt[{6}]{3}}}}$	${\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}}$
${\displaystyle {\tfrac {7}{12}}\pi _{3}}$	${\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}}$	${\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}}$

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

${\displaystyle {\begin{aligned}\operatorname {cm} (u+v)&={\frac {\operatorname {sm} u\,\operatorname {cm} u-\operatorname {sm} v\,\operatorname {cm} v}{\operatorname {sm} u\,\operatorname {cm} ^{2}v-\operatorname {cm} ^{2}u\,\operatorname {sm} v}}\\[8mu]\operatorname {cm} (u-v)&={\frac {\operatorname {cm} ^{2}u\,\operatorname {cm} v-\operatorname {sm} u\,\operatorname {sm} ^{2}v}{\operatorname {cm} u\,\operatorname {cm} ^{2}v-\operatorname {sm} ^{2}u\,\operatorname {sm} v}}\\[8mu]\operatorname {sm} (u+v)&={\frac {\operatorname {sm} ^{2}u\,\operatorname {cm} v-\operatorname {cm} u\,\operatorname {sm} ^{2}v}{\operatorname {sm} u\,\operatorname {cm} ^{2}v-\operatorname {cm} ^{2}u\,\operatorname {sm} v}}\\[8mu]\operatorname {sm} (u-v)&={\frac {\operatorname {sm} u\,\operatorname {cm} u-\operatorname {sm} v\,\operatorname {cm} v}{\operatorname {cm} u\,\operatorname {cm} ^{2}v-\operatorname {sm} ^{2}u\,\operatorname {sm} v}}\end{aligned}}}$

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}}\\[4mu]&={\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}}\\[4mu]&\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}}\\[8mu]\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}}\\[4mu]&={\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}}\\[4mu]&\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}}}$

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

${\displaystyle {\begin{aligned}\operatorname {cm} 2u&={\frac {\operatorname {cm} ^{3}u-\operatorname {sm} ^{3}u}{\operatorname {cm} u(1+\operatorname {sm} ^{3}u)}}={\frac {2\operatorname {cm} ^{3}u-1}{2\operatorname {cm} u-\operatorname {cm} ^{4}u}},\\[5mu]\operatorname {sm} 2u&={\frac {\operatorname {sm} u(1+\operatorname {cm} ^{3}u)}{\operatorname {cm} u(1+\operatorname {sm} ^{3}u)}}={\frac {2\operatorname {sm} u-\operatorname {sm} ^{4}u}{2\operatorname {cm} u-\operatorname {cm} ^{4}u}},\\[5mu]\operatorname {cm} 3u&={\frac {\operatorname {cm} ^{9}u-6\operatorname {cm} ^{6}u+3\operatorname {cm} ^{3}u+1}{\operatorname {cm} ^{9}u+3\operatorname {cm} ^{6}u-6\operatorname {cm} ^{3}u+1}},\\[5mu]\operatorname {sm} 3u&={\frac {3\operatorname {sm} u\,\operatorname {cm} u(\operatorname {sm} ^{3}u\,\operatorname {cm} ^{3}u-1)}{\operatorname {cm} ^{9}u+3\operatorname {cm} ^{6}u-6\operatorname {cm} ^{3}u+1}}.\end{aligned}}}$

The ${\displaystyle \operatorname {cm} }$ function satisfies the identities $${\displaystyle {\begin{aligned}\operatorname {cm} {\tfrac {2}{9}}\pi _{3}&=-\operatorname {cm} {\tfrac {1}{9}}\pi _{3}\,\operatorname {cm} {\tfrac {4}{9}}\pi _{3},\\[5mu]\operatorname {cm} {\tfrac {1}{4}}\pi _{3}&=\operatorname {cl} {\tfrac {1}{3}}\varpi ,\end{aligned}}}$$

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

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

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

whose coefficients satisfy the recurrence ${\displaystyle c_{0}=s_{0}=1,}$^[10]

${\displaystyle {\begin{aligned}c_{n}&=-{\frac {1}{3n}}\sum _{k=0}^{n-1}s_{k}s_{n-1-k}\\[4mu]s_{n}&={\frac {1}{3n+1}}\sum _{k=0}^{n}c_{k}c_{n-k}\end{aligned}}}$

These recurrences result in:^[11]

${\displaystyle {\begin{aligned}\operatorname {cm} z&=1-{\frac {1}{3}}z^{3}+{\frac {1}{18}}z^{6}-{\frac {23}{2268}}z^{9}+{\frac {25}{13608}}z^{12}-{\frac {619}{1857492}}z^{15}+\cdots \\[8mu]\operatorname {sm} z&=z-{\frac {1}{6}}z^{4}+{\frac {2}{63}}z^{7}-{\frac {13}{2268}}z^{10}+{\frac {23}{22113}}z^{13}-{\frac {2803}{14859936}}z^{16}+\cdots \end{aligned}}}$

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:^[12]

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

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:^[13]

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

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:

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

The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley.^[14] 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:

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

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.^[15]

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

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

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

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]

• 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

• 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/