Dixon elliptic functions

The functions sm and cm can be defined as the solutions to the initial value problem:Dixon (1890), Van Fossen Conrad & Flajolet (2005), Robinson (2019).

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

Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:The mapping for a general regular polygon is described in Schwarz (1869).

:$z\; =\; \backslash int\_0^\{\backslash operatorname\{sm\}\; z\}\; \backslash frac\{dw\}\{(1\; -\; w^3)^\{2/3\}\}\; =\; \backslash int\_\{\backslash operatorname\{cm\}\; z\}^1\; \backslash frac\{dw\}\{(1\; -\; w^3)^\{2/3\}\}$

which can also be expressed using the hypergeometric function:van Fossen Conrad & Flajolet (2005) p. 6.

:$\backslash operatorname\{sm\}^\{-1\}(z)\; =\; z\backslash ;\; \{\}\_2F\_1\backslash bigl(\backslash tfrac13,\; \backslash tfrac23;\; \backslash tfrac43;\; z^3\backslash bigr)$

File:Dixon cm, sm on the cubic Fermat curve.png

Both sm and cm have a period along the real axis of $\backslash pi\_3\; =\; \backslash Beta\backslash bigl(\; \backslash tfrac13,\; \backslash tfrac13\backslash bigr)\; =\; \backslash tfrac\{\backslash sqrt\{3\}\}\{2\backslash pi\}\backslash Gamma^3\backslash bigl(\backslash tfrac\{1\}\{3\}\backslash bigr)\backslash approx\; 5.29991625$ with $\backslash Beta$ the beta function and $\backslash Gamma$ the gamma function:Dillner (1873) calls the period $3w$. Dixon (1890) calls it $3\backslash lambda$; Adams (1925) and Robinson (2019) each call it $3K$. Van Fossen Conrad & Flajolet (2005) call it $\backslash pi\_3$. Also see OEIS [https://oeis.org/A197374 A197374].

:

They satisfy the identity $\backslash operatorname\{cm\}^3\; z\; +\; \backslash operatorname\{sm\}^3\; z\; =\; 1$. The parametric function $t\; \backslash mapsto\; (\backslash operatorname\{cm\}\; t,\backslash ,\; \backslash operatorname\{sm\}\; t),$ $t\; \backslash in\; \backslash bigl[\{-\backslash tfrac13\}\backslash pi\_3,\; \backslash tfrac23\backslash pi\_3\backslash bigr]$ parametrizes the cubic Fermat curve $x^3\; +\; y^3\; =\; 1,$ with $\backslash tfrac12\; t$ representing the signed area lying between the segment from the origin to $(1,\backslash ,\; 0)$, the segment from the origin to $(\backslash operatorname\{cm\}\; t,\backslash ,\; \backslash operatorname\{sm\}\; t)$, 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.Dixon (1890), Van Fossen Conrad & Flajolet (2005) To see why, apply Green's theorem:

:$A\; =\; \backslash tfrac\; 12\; \backslash int\_0^t\; (x\backslash mathop\{dy\}\; -y\backslash mathop\{dx\})\; =\; \backslash tfrac\; 12\; \backslash int\_0^t\; (\backslash operatorname\{cm\}^3\; t\; +\; \backslash operatorname\{sm\}^3\; t)\backslash mathop\{dt\}\; =\; \backslash tfrac\; 12\; \backslash int\_0^t\; dt\; =\; \backslash tfrac12\; t.$

Notice that the area between the $x\; +\; y\; =\; 0$ and $x^3\; +\; y^3\; =\; 1$ can be broken into three pieces, each of area $\backslash tfrac16\backslash pi\_3$:

:$\backslash begin\{aligned\}$

\tfrac12\pi_3 &= \int_{-\infty}^\infty \bigl((1 - x^3)^{1/3} + x\bigr)\mathop{dx} \\[8mu]

\tfrac16\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}

File:Sm in the complex plane.svg of $\backslash operatorname\{sm\}z$ goes from $-\backslash pi$ to $\backslash pi$, the colors go through cyan, blue ($\backslash operatorname\{Arg\}\backslash approx\; -\backslash pi/2$), magneta, red ($\backslash operatorname\{Arg\}\backslash approx\; 0$), orange, yellow ($\backslash operatorname\{Arg\}\backslash approx\backslash pi/2$), green, and back to cyan ($\backslash operatorname\{Arg\}\backslash approx\backslash pi$).]]

The function $\backslash operatorname\{sm\}\; z$ has zeros at the complex-valued points $z\; =\; \backslash tfrac1\backslash sqrt\{3\}\backslash pi\_3i(a\; +\; b\backslash omega)$ for any integers $a$ and $b$, where $\backslash omega$ is a cube root of unity, $\backslash omega\; =\; \backslash exp\; \backslash tfrac23\; i\; \backslash pi\; =\; -\backslash tfrac12\; +\; \backslash tfrac\backslash sqrt\{3\}2i$ (that is, $a\; +\; b\backslash omega$ is an Eisenstein integer). The function $\backslash operatorname\{cm\}\; z$ has zeros at the complex-valued points $z\; =\; \backslash tfrac13\backslash pi\_3\; +\; \backslash tfrac1\backslash sqrt\{3\}\backslash pi\_3i(a\; +\; b\backslash omega)$. Both functions have poles at the complex-valued points $z\; =\; -\backslash tfrac13\backslash pi\_3\; +\; \backslash tfrac1\backslash sqrt\{3\}\backslash pi\_3i(a\; +\; b\backslash omega)$.

On the real line, $\backslash operatorname\{sm\}x=0\backslash leftrightarrow\; x\backslash in\backslash pi\_3\backslash mathbb\{Z\}$, which is analogous to $\backslash sin\; x=0\backslash leftrightarrow\; x\backslash in\backslash pi\backslash mathbb\{Z\}$.

Both {{math|cm}} and {{math|sm}} commute with complex conjugation,

: $\backslash begin\{align\}$

\operatorname{cm} \bar{z} &= \overline{\operatorname{cm} z}, \\

\operatorname{sm} \bar{z} &= \overline{\operatorname{sm} z}.

\end{align}

Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), the Dixon function {{math|cm}} is invariant under $\backslash tfrac13$ turn rotations of the complex plane, and $\backslash tfrac13$ turn rotations of the domain of {{math|sm}} cause $\backslash tfrac13$ turn rotations of the codomain:

:$\backslash begin\{align\}$

\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{align}

Each Dixon elliptic function is invariant under translations by the Eisenstein integers $a\; +\; b\backslash omega$ scaled by $\backslash pi\_3,$

: $\backslash begin\{align\}$

\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{align}

Negation of each of {{math|cm}} and {{math|sm}} is equivalent to a $\backslash tfrac13\backslash pi\_3$ translation of the other,

:$\backslash begin\{align\}$

\operatorname{cm}(-z) &= \frac{1}{\operatorname{cm} z} = \operatorname{sm} \bigl(z + \tfrac13\pi_3\bigr), \\

\operatorname{sm}(-z) &= -\frac{\operatorname{sm} z}{\operatorname{cm} z} = \frac{1}{\operatorname{sm} \bigl(z - \tfrac13\pi_3\bigr)} = \operatorname{cm} \bigl(z + \tfrac13\pi_3\bigr).

\end{align}

For $n\; \backslash in\; \backslash mathbb\; \backslash \{0,\; 1,\; 2\backslash \},$ translations by $\backslash tfrac13\backslash pi\_3\backslash omega$ give

: $\backslash begin\{align\}$

\operatorname{cm}\bigl(z+\tfrac13\omega^n\pi_3\bigr) &= \omega^{2n}\frac{-\operatorname{sm} z}{\operatorname{cm} z}, \\

\operatorname{sm}\bigl(z+\tfrac13\omega^n\pi_3\bigr) &= \omega^n\frac{1}{\operatorname{cm} z}.

\end{align}

The Dixon elliptic functions satisfy the argument sum and difference identities:Dixon (1890), Adams (1925)

:$\backslash begin\{aligned\}$

\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:{{citation needed|date=January 2023}}

:$\backslash begin\{aligned\}$

\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:Dixon (1890), [https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify=%7B%22pages%22%3A%5B197%5D%7D p. 185–186]. Robinson (2019).

:$\backslash begin\{align\}$

\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}^3u\, \operatorname{cm}^3u - 1)}

{ \operatorname{cm}^9 u + 3\operatorname{cm}^6 u - 6\operatorname{cm}^3 u + 1}.

\end{align}

The $\backslash operatorname\{cm\}$ function satisfies the identities

$$\backslash begin\{align\}$$

\operatorname{cm}\tfrac29\pi_3 &= -\operatorname{cm}\tfrac19 \pi_3\, \operatorname{cm}\tfrac49\pi_3, \\[5mu]

\operatorname{cm}\tfrac14\pi_3 &= \operatorname{cl}\tfrac13\varpi,

\end{align}

where $\backslash operatorname\{cl\}$ is lemniscate cosine and $\backslash varpi$ is Lemniscate constant.{{citation needed|date=June 2023}}

The {{math|cm}} and {{math|sm}} functions can be approximated for by the Taylor series

:$\backslash begin\{aligned\}$

\operatorname{cm} z &= c_0 + c_1z^3 + c_2z^6 + c_3z^{9} + \cdots + c_nz^{3n} + \cdots \\[4mu]

\operatorname{sm} z &= s_0z + s_1z^4 + s_2z^7 + s_3z^{10} + \cdots + s_nz^{3n+1} + \cdots

\end{aligned}

whose coefficients satisfy the recurrence $c\_0\; =\; s\_0\; =\; 1,$Adams (1925)

:$\backslash begin\{aligned\}$

c_n &= -\frac{1}{3n}\sum_{k=0}^{n-1} s_ks_{n-1-k} \\[4mu]

s_n &= \frac{1}{3n + 1}\sum_{k=0}^n c_kc_{n-k}

\end{aligned}

These recurrences result in:van Fossen Conrad & Flajolet (2005). Also see OEIS [https://oeis.org/A104133 A104133], [https://oeis.org/A104134 A104134].

:$\backslash begin\{aligned\}$

\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}

File:Weierstrass cubic curve related to the Dixon elliptic functions.png $y^2\; =\; 4x^3\; -\; \backslash tfrac1\{27\}$ for the Weierstrass ℘-function $z\; \backslash mapsto\; \backslash wp\backslash bigl(z;\; 0,\; \backslash tfrac1\{27\}\backslash bigr)$ related to the Dixon elliptic functions.]]

The equianharmonic Weierstrass elliptic function $\backslash wp(z)\; =\; \backslash wp\backslash bigl(z;\; 0,\; \backslash tfrac1\{27\}\backslash bigr),$ with lattice $\backslash Lambda\; =\; \backslash pi\_3\backslash mathbb\{Z\}\; \backslash oplus\; \backslash pi\_3\backslash omega\backslash mathbb\{Z\}$ a scaling of the Eisenstein integers, can be defined as:Reinhardt & Walker (2010)

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

The function $\backslash wp(z)$ solves the differential equation:

:$\backslash wp\text{'}(z)^2\; =\; 4\backslash wp(z)^3\; -\; \backslash tfrac1\{27\}$

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

:$z\; =\; \backslash int\_\backslash infty^\{\backslash wp(z)\}\; \backslash frac\{dw\}\{\backslash sqrt\{4w^3\; -\; \backslash tfrac1\{27\}\}\}$

In terms of $\backslash wp(z)$, the Dixon elliptic functions can be written:Chapling (2018), Robinson (2019). Adams (1925) instead expresses the Dixon elliptic functions in terms of the Weierstrass elliptic function $\backslash wp(z;\; 0,\; -1).$

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

Likewise, the Weierstrass elliptic function $\backslash wp(z)\; =\; \backslash wp\backslash bigl(z;\; 0,\; \backslash tfrac1\{27\}\backslash bigr)$ can be written in terms of Dixon elliptic functions:

: $\backslash wp\text{'}(z)\; =\; \backslash frac\{\backslash operatorname\{cm\}\; z\; +\; 1\}\{3(\backslash operatorname\{cm\}\; z\; -\; 1)\},\backslash \; \backslash wp(z)\; =\; \backslash frac\{-\backslash operatorname\{sm\}\; z\}\{3(\backslash operatorname\{cm\}\; z\; -\; 1)\}$

The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley.van Fossen Conrad & Flajolet (2005), p.38 Let $k\; =\; e^\{5\; i\; \backslash pi\; /\; 6\}$, $\backslash theta\; =\; 3^\{\backslash frac\{1\}\{4\}\}\; e^\{5\; i\; \backslash pi\; /\; 12\}$, $s\; =\; \backslash operatorname\; \{sn\}(u,k)$, $c\; =\; \backslash operatorname\; \{cn\}(u,k)$, and $d\; =\; \backslash operatorname\; \{dn\}(u,k)$. Then, let

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

Finally, the Dixon elliptic functions are as so:

: $\backslash operatorname\; \{sm\}(z)\; =\; \backslash xi\; \backslash left(\backslash frac\{z\; +\; \backslash pi\_3/6\}\{2^\{1/3\}\; \backslash theta\}\backslash right),$ $\backslash operatorname\; \{cm\}(z)\; =\; \backslash eta\; \backslash left(\backslash frac\{z\; +\; \backslash pi\_3/6\}\{2^\{1/3\}\; \backslash theta\}\backslash right).$

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.Lundberg (1879), Grammel (1948), Shelupsky (1959), Burgoyne (1964), Gambini, Nicoletti, & Ritelli (2021).

For example, defining $\backslash pi\_n\; =\; \backslash Beta\backslash bigl(\backslash tfrac1n,\; \backslash tfrac1n\backslash bigr)$ and $\backslash sin\_n\; z,\backslash ,\backslash cos\_n\; z$ the inverses of an integral:

:$z\; =\; \backslash int\_0^\{\backslash sin\_n\; z\}\; \backslash frac\{dw\}\{(1\; -\; w^n)^\{(n-1)/n\}\}\; =\; \backslash int\_\{\backslash cos\_n\; z\}^1\; \backslash frac\{dw\}\{(1\; -\; w^n)^\{(n-1)/n\}\}$

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

:$\backslash int\_0^1\; (1\; -\; x^n)^\{1/n\}\; \backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{\backslash pi\_n\}\{2n\}.$

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

File:Conformal map projection from globe to octahedron.png

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.Adams (1925), Cox (1935), Magis (1938), Lee (1973), Lee (1976), McIlroy (2011), Chapling (2016).

• 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

{{reflist|30em}}

{{refbegin|30em}}

• 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 $\backslash int\; dx\; /\; (1\; -\; x^3)\{\}^\{2/3\}$ 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 {{nowrap|{{math|x{{sup|3}} + y{{sup|3}} − 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/
• {{cite journal

| first = A. C.

| last = Dixon

| authorlink = Alfred Cardew Dixon

| title = On the doubly periodic functions arising out of the curve {{nowrap|{{math|x{{sup|3}} + y{{sup|3}} − 3αxy {{=}} 1}}}}

| journal = Quarterly Journal of Pure and Applied Mathematics

| year = 1890

| volume = XXIV

| pages = 167–233

| url = https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify={%22pages%22:%5b179%5d} }}

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| arxiv = math/0507268

| date = 2005

| journal = Séminaire Lotharingien de Combinatoire

| mr = 2223029

| page = Art. B54g, 44

| title = The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion

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• 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
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• 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) [https://archive.org/details/conformalproject0000leel 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 [https://www.utpjournals.press/toc/cart/13/1 The Canadian Cartographer 13].
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{{refend}}

• 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 $x^3+y^3=z^3$, 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/

Category:Complex analysis

Category:Elliptic functions