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Dixon elliptic function specific values

Dixon elliptic functions, are Elliptic functions which parametrize $x^{3}+y^{3}=1$ Fermat curve and are useful for Conformal map projections from Sphere to Triangle-related shapes. It is known that $\forall n,m,k\in \mathbb {Z}$ $\operatorname {cm} ({\frac {n\pi _{3}+m\omega \pi _{3}}{k}})\in \mathbb {A} \cup \{\infty \}$ and $\operatorname {sm} ({\frac {n\pi _{3}+m\omega \pi _{3}}{k}})\in \mathbb {A} \cup \{\infty \}$ where $\mathbb {A}$ denotes set of all Algebraic numbers also $\operatorname {cm} ({\frac {k\pi _{3}}{2^{n}3^{m}}})\in \mathbb {M} \cup \{\infty \}$ and $\operatorname {sm} ({\frac {k\pi _{3}}{2^{n}3^{m}}})\in \mathbb {M} \cup \{\infty \}$ where $\mathbb {M}$ denotes set of all Origami-constructibles. Where $\omega =e^{\frac {2\pi i}{3}}$

Simple real values

$z$	$\operatorname {cm} z$	$\operatorname {sm} z$
${-{\tfrac {1}{3}}}\pi _{3}$	$\infty$	$\infty$
${-{\tfrac {1}{4}}}\pi _{3}$	${\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}$	${\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}$
${-{\tfrac {1}{6}}}\pi _{3}$	${\sqrt[{3}]{2}}$	$-1$
$-{\tfrac {1}{12}}\pi _{3}$	${\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}$	${\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}$
$0$	$1$	$0$
${\tfrac {1}{12}}\pi _{3}$	${\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}$	${\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}$
${\tfrac {1}{6}}\pi _{3}$	$1{\big /}{\sqrt[{3}]{2}}$	$1{\big /}{\sqrt[{3}]{2}}$
${\tfrac {1}{4}}\pi _{3}$	${\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}$	${\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}$
${\tfrac {1}{3}}\pi _{3}$	$0$	$1$
${\tfrac {5}{12}}\pi _{3}$	${\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}$	${\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}$
${\tfrac {1}{2}}\pi _{3}$	$-1$	${\sqrt[{3}]{2}}$
${\tfrac {7}{12}}\pi _{3}$	${\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}$	${\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}$
${\tfrac {2}{3}}\pi _{3}$	$\infty$	$\infty$

Complex specific values

$\operatorname {cm} (a+b\omega )$	$b$	$0$	${\tfrac {1}{6}}\pi _{3}$	${\tfrac {1}{3}}\pi _{3}$	${\tfrac {1}{2}}\pi _{3}$	${\tfrac {2}{3}}\pi _{3}$	${\tfrac {5}{6}}\pi _{3}$	$\pi _{3}$
$a$
$0$		$1$	$1{\big /}{\sqrt[{3}]{2}}$	$0$	$-1$	$\infty$	${\sqrt[{3}]{2}}$	$1$
${\tfrac {1}{6}}\pi _{3}$		$1{\big /}{\sqrt[{3}]{2}}$	${\sqrt[{3}]{2}}$	$-\omega ^{2}$	$\omega {\big /}{\sqrt[{3}]{2}}$	$\omega {\sqrt[{3}]{2}}$	$-\omega ^{2}$	$1{\big /}{\sqrt[{3}]{2}}$
${\tfrac {1}{3}}\pi _{3}$		$0$	$-\omega$	$\infty$	$\omega {\sqrt[{3}]{2}}$	$\omega$	$\omega {\big /}{\sqrt[{3}]{2}}$	$0$
${\tfrac {1}{2}}\pi _{3}$		$-1$	$\omega ^{2}{\big /}{\sqrt[{3}]{2}}$	$\omega ^{2}{\sqrt[{3}]{2}}$	$-1$	$\omega {\big /}{\sqrt[{3}]{2}}$	$\omega {\sqrt[{3}]{2}}$	$-1$
${\tfrac {2}{3}}\pi _{3}$		$\infty$	$\omega ^{2}{\sqrt[{3}]{2}}$	$\omega ^{2}$	$\omega ^{2}{\big /}{\sqrt[{3}]{2}}$	$0$	$-\omega ^{2}$	$\infty$
${\tfrac {5}{6}}\pi _{3}$		${\sqrt[{3}]{2}}$	$-\omega$	$\omega ^{2}{\big /}{\sqrt[{3}]{2}}$	$\omega ^{2}{\sqrt[{3}]{2}}$	$-\omega$	$1{\big /}{\sqrt[{3}]{2}}$	${\sqrt[{3}]{2}}$
$\pi _{3}$		$1$	$1{\big /}{\sqrt[{3}]{2}}$	$0$	$-1$	$\infty$	${\sqrt[{3}]{2}}$	$1$

$\operatorname {sm} (a+b\omega )$	$b$	$0$	${\tfrac {1}{6}}\pi _{3}$	${\tfrac {1}{3}}\pi _{3}$	${\tfrac {1}{2}}\pi _{3}$	${\tfrac {2}{3}}\pi _{3}$	${\tfrac {5}{6}}\pi _{3}$	$\pi _{3}$
$a$
$0$		$0$	$\omega {\big /}{\sqrt[{3}]{2}}$	$\omega$	$\omega {\sqrt[{3}]{2}}$	$\infty$	$-\omega$	$0$
${\tfrac {1}{6}}\pi _{3}$		$1{\big /}{\sqrt[{3}]{2}}$	$-\omega ^{2}$	$\omega {\sqrt[{3}]{2}}$	$\omega {\big /}{\sqrt[{3}]{2}}$	$-\omega ^{2}$	${\sqrt[{3}]{2}}$	$1{\big /}{\sqrt[{3}]{2}}$
${\tfrac {1}{3}}\pi _{3}$		$1$	${\sqrt[{3}]{2}}$	$\infty$	$-1$	$0$	$1{\big /}{\sqrt[{3}]{2}}$	$1$
${\tfrac {1}{2}}\pi _{3}$		${\sqrt[{3}]{2}}$	$1{\big /}{\sqrt[{3}]{2}}$	$-\omega$	$\omega ^{2}{\sqrt[{3}]{2}}$	$\omega ^{2}{\big /}{\sqrt[{3}]{2}}$	$-\omega$	${\sqrt[{3}]{2}}$
${\tfrac {2}{3}}\pi _{3}$		$\infty$	$-\omega ^{2}$	$0$	$\omega ^{2}{\big /}{\sqrt[{3}]{2}}$	$\omega ^{2}$	$\omega ^{2}{\sqrt[{3}]{2}}$	$\infty$
${\tfrac {5}{6}}\pi _{3}$		$-1$	$\omega {\sqrt[{3}]{2}}$	$\omega {\big /}{\sqrt[{3}]{2}}$	$-1$	$\omega ^{2}{\sqrt[{3}]{2}}$	$\omega ^{2}{\big /}{\sqrt[{3}]{2}}$	$-1$
$\pi _{3}$		$0$	$\omega {\big /}{\sqrt[{3}]{2}}$	$\omega$	$\omega {\sqrt[{3}]{2}}$	$\infty$	$-\omega$	$0$

Deriviation methods

For one deriviation method, we substitute $x$ and $y\omega$ in sum identities, and make use of reflexion identities $\operatorname {cm} y\omega =\operatorname {cm} y$ and $\operatorname {sm} y\omega =\omega \operatorname {sm} y$ to get:

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

For example:

${\begin{aligned}\operatorname {cm} ({\tfrac {1}{6}}\pi _{3}+\omega {\tfrac {1}{3}}\pi _{3})&={\frac {1{\big /}{\sqrt[{3}]{4}}}{-\omega \,{\big /}{\sqrt[{3}]{4}}}}=-\omega ^{2}\end{aligned}}$

Another way to deriviate specific values, is to make use of multiple-argument formulas:

For example, to calculate $\operatorname {cm} ({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})$, we use cm duplication formula,

${\begin{aligned}\operatorname {cm} 2u&={\frac {2\operatorname {cm} ^{3}u-1}{2\operatorname {cm} u-\operatorname {cm} ^{4}u}},\end{aligned}}$

${\begin{aligned}\operatorname {cm} ({\tfrac {1}{2}}\pi _{3}+\omega \pi _{3})&=\operatorname {cm} ({\tfrac {1}{2}}\pi _{3})=-1={\frac {2\operatorname {cm} ^{3}({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})-1}{2\operatorname {cm} ({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})-\operatorname {cm} ^{4}({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})}},\end{aligned}}$

Equation $x^{4}-2x^{3}-2x+1=0$ has 4 roots:

${\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}$

${\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}$

${\frac {1-{\sqrt {3}}-i{\sqrt {2{\sqrt {3}}}}}{2}}$

${\frac {1-{\sqrt {3}}+i{\sqrt {2{\sqrt {3}}}}}{2}}$

By looking at complex cm domain coloring, we can deduct that $\operatorname {cm} ({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})$ is non-real with positive argument less than $\pi$. A complex number has positive argument less than $\pi$ if and only if it's imaginary part is positive, so:

$\operatorname {cm} ({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})={\frac {1-{\sqrt {3}}+i{\sqrt {2{\sqrt {3}}}}}{2}}$

Generalized Fermat curve trigonometric functions

In mathematics, Generalised Fermat curve trigonometric functions are complex functions $\operatorname {cos_{n}} z,\,\operatorname {sin_{n}} z$ which real values parametrize curve $x^{n}+y^{n}=1$. That's why these functions satisfy the identity $\operatorname {cos_{n}} ^{n}z+\operatorname {sin_{n}} ^{n}z=1$. They are generalizations of regular Trigonometric functions which are the case when $n=2$. Generalization of $\pi$ for other Fermat curves is: $\pi _{n}=\mathrm {B} {\bigl (}{\tfrac {1}{n}},{\tfrac {1}{n}}{\bigr )}={\frac {\Gamma ^{2}({\frac {1}{n}})}{\Gamma ({\frac {2}{n}})}}$.

Parametrization of Fermat curves

$\sin _{n}z,\,\cos _{n}z$ are inverses of these integrals:

$z=\int _{0}^{\sin _{n}z}(1-t^{n})^{\frac {1-n}{n}}dt=\int _{\cos _{n}z}^{1}(1-t^{n})^{\frac {1-n}{n}}dt$

They also parametrize $x^{n}+y^{n}=1$, in a way that the signed area lying between the segment from the origin to $(\cos _{n}z,\,\sin _{n}z)$ is ${\tfrac {1}{2}}z$ for $z\in [0,{\frac {\pi _{n}}{n}}]$.

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

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

$m$	$\operatorname {cos_{n}} m$	$\operatorname {sin_{n}} m$
$0$	$1$	$0$
${\frac {\pi _{n}}{2n}}$	$1/{\sqrt[{n}]{2}}$	$1/{\sqrt[{n}]{2}}$
${\frac {\pi _{n}}{n}}$	$0$	$1$

Trigonometric functions

In case when $n=2$, we get Trigonometric functions $\operatorname {cos}$ and $\operatorname {sin}$ which satisfy $\cos ^{2}x+\sin ^{2}x=1$ and parametrize Unit circle.

Reflection identities

$\operatorname {cos} (z+\pi )=-\operatorname {cos} (z)$

$\operatorname {sin} (z+\pi )=-\operatorname {sin} (z)$

$\operatorname {cos} (\pi -z)=-\operatorname {cos} (z)$

$\operatorname {sin} (\pi -z)=\operatorname {sin} (z)$

$\operatorname {cos} (-z)=\operatorname {cos} (z)$

$\operatorname {sin} (-z)=-\operatorname {sin} (z)$

$\operatorname {cos} (z+2\pi )=\operatorname {cos} (z)$

$\operatorname {sin} (z+2\pi )=\operatorname {sin} (z)$

$\operatorname {cos} ({\frac {\pi }{2}}-z)=\operatorname {sin} (z)$

$\operatorname {sin} ({\frac {\pi }{2}}-z)=\operatorname {cos} (z)$

Specific Values

$n$	$\cos(n)$	$\sin(n)$
$0$	$1$	$0$
${\frac {\pi }{12}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$
${\frac {\pi }{6}}$	${\frac {\sqrt {3}}{2}}$	${\frac {1}{2}}$
${\frac {\pi }{4}}$	${\frac {\sqrt {2}}{2}}$	${\frac {\sqrt {2}}{2}}$
${\frac {\pi }{3}}$	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$
${\frac {5\pi }{12}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$
${\frac {\pi }{2}}$	$0$	$1$

Multiple Argument identities

$\cos(2n)=2\cos ^{2}n-1$

$\sin(2n)=2\sin n\cos n$

$\cos(3n)=4\cos ^{3}n-3\cos n$

$\sin(3n)=3\sin n-4\sin ^{3}n$

Sum and Difference identities

${\begin{aligned}\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y\end{aligned}}$

Derivatives

$\operatorname {cos'} (z)=-\operatorname {sin} (z)$

$\operatorname {sin'} (z)=\operatorname {cos} (z)$

Dixon elliptic functions

In case when $n=3$, we get Dixon elliptic functions $\operatorname {cm}$ and $\operatorname {sm}$ which satisfy $\operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1$ with period of $\pi _{3}$, which parametrize the cubic Fermat curve $x^{3}+y^{3}=1$.

Let $\omega =\exp {\tfrac {2}{3}}i\pi =-{\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i$.

Reflection identities

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

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

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

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

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

Specific Values

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

Multiple Argument identities

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

Sum and Difference identities

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

Derivatives

$\operatorname {cm'} (z)=-\operatorname {sm} ^{2}(z)$

$\operatorname {sm'} (z)=\operatorname {cm} ^{2}(z)$

Quartic Trigonometric functions

In case when $n=4$, we get $\operatorname {cos_{4}}$ and $\operatorname {sin_{4}}$ which satisfy $\operatorname {cos_{4}} ^{4}z+\operatorname {sin_{4}} ^{4}z=1$ with period of $\pi _{4}=2{\sqrt {2}}\varpi$, which parametrize the quartic Fermat curve $x^{4}+y^{4}=1$. Unlike previous cases, they are not meromorphic, but their squares and ratios are. They are related to Lemniscate elliptic functions by ${\frac {\operatorname {sin_{4}} n}{\operatorname {cos_{4}} n}}=\operatorname {slh} n$, where $\operatorname {slh}$ is hyperbolic lemiscate sine which is related to regular lemniscate functions by:$\operatorname {slh} ({\sqrt {2}}z)={\frac {1-i}{\sqrt {2}}}\operatorname {sl} ((1+i)z)={\frac {\operatorname {sl} ({\sqrt[{4}]{-4}}z)}{\sqrt[{4}]{-1}}}={\frac {(1+\operatorname {cl} ^{2}z)\operatorname {sl} z}{{\sqrt {2}}\operatorname {cl} z}}$

Specific Values

$n$	$\operatorname {cos_{4}} n$	$\operatorname {sin_{4}} n$
$0$	$1$	$0$
${\tfrac {1}{16}}\pi _{4}$	${\sqrt[{4}]{\frac {1+{\sqrt {2{\sqrt {2}}-2}}}{2}}}$	${\sqrt[{4}]{\frac {1-{\sqrt {2{\sqrt {2}}-2}}}{2}}}$
${\tfrac {1}{12}}\pi _{4}$	${\sqrt[{8}]{\frac {3}{4}}}$	${\sqrt {\frac {{\sqrt {3}}-1}{2}}}$
${\tfrac {1}{8}}\pi _{4}$	$1{\big /}{\sqrt[{4}]{2}}$	$1{\big /}{\sqrt[{4}]{2}}$
${\tfrac {1}{4}}\pi _{4}$	$0$	$1$
${\tfrac {1}{2}}\pi _{4}$	$-1$	$0$
${\tfrac {3}{4}}\pi _{4}$	$0$	$-1$
$\pi _{4}$	$1$	$0$