User:Great Cosine/sandbox

= 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 $$ \operatorname{cm}(\frac {k \pi_3} {2^n3^m}) \in \mathbb M\cup\{\infty\}$$ and $$ \operatorname{sm}(\frac {k \pi_3} {2^n3^m}) \in \mathbb M\cup\{\infty\}$$ where $$ \mathbb M$$ denotes set of all Origami-constructibles. Where $$ \omega=e^\frac{2\pi i}{3}$$

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}({\tfrac16}\pi_3 + \omega {\tfrac13}\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 $$\operatorname{cm}({\tfrac14}\pi_3 + \omega {\tfrac12}\pi_3)$$, we use cm duplication formula,


 * $$\begin{align}

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

\operatorname{cm}({\tfrac12}\pi_3 + \omega \pi_3) &= \operatorname{cm}({\tfrac12}\pi_3)=-1 =\frac { 2\operatorname{cm}^3({\tfrac14}\pi_3 + \omega {\tfrac12}\pi_3) - 1} { 2\operatorname{cm}({\tfrac14}\pi_3 + \omega {\tfrac12}\pi_3) - \operatorname{cm}^4({\tfrac14}\pi_3 + \omega {\tfrac12}\pi_3)}, \end{align}$$

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}({\tfrac14}\pi_3 + \omega {\tfrac12}\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}({\tfrac14}\pi_3 + \omega {\tfrac12}\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: $$\pi_n = \Beta\bigl(\tfrac1n, \tfrac1n\bigr) = \frac{\Gamma^2(\frac{1}{n})}{\Gamma(\frac{2}{n})}$$.

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 $$\tfrac12 z$$ for $$z \in [0, \frac{\pi_n}{n}]$$.

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


 * $$\int_0^{1} (1 - x^n)^{1/n}\mathop{dx} = \frac{\pi_n}{2n}$$.

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

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

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

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

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

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

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

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

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

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

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

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

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 $$\omega = \exp \tfrac23 i \pi = -\tfrac12 + \tfrac\sqrt{3}2i$$.

Reflection identities

 * $$\begin{align}

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

\operatorname{cm}\bigl(z+\tfrac13\omega\pi_3\bigr) &= \omega^{2}\frac{-\operatorname{sm} z}{\operatorname{cm} z}, \\ \operatorname{sm}\bigl(z+\tfrac13\omega\pi_3\bigr) &= \omega\frac{1}{\operatorname{cm} z}. \end{align}$$

Multiple Argument identities
$$\begin{align}

\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}^3n\, \operatorname{cm}^3n - 1)} { \operatorname{cm}^9 n + 3\operatorname{cm}^6 n - 6\operatorname{cm}^3 n + 1}.

\end{align}$$

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 $$\pi_4=2\sqrt2\varpi$$, 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:$$\operatorname{slh}(\sqrt2 z) = \frac{1-i}{\sqrt2} \operatorname{sl}((1+i)z) = \frac{\operatorname{sl}(\sqrt[4]{-4}z) }{ \sqrt[4]{-1} } = \frac{(1+\operatorname{cl}^2 z)\operatorname{sl} z}{\sqrt2\operatorname{cl} z} $$