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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 [math]\displaystyle{ \operatorname{cm}^3 z + \operatorname{sm}^3 z = 1 }[/math], as real functions they parametrize the cubic Fermat curve [math]\displaystyle{ x^3 + y^3 = 1 }[/math], just as the trigonometric functions sine and cosine parametrize the unit circle [math]\displaystyle{ x^2 + y^2 = 1 }[/math].
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]
Definition
The functions sm and cm can be defined as the solutions to the initial value problem:[2]
- [math]\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 }[/math]
Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:[3]
- [math]\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}} }[/math]
which can also be expressed using the hypergeometric function:[4]
- [math]\displaystyle{ \operatorname{sm}^{-1}(z) = z\; {}_2F_1\bigl(\tfrac13, \tfrac23; \tfrac43; z^3\bigr) }[/math]
Parametrization of the cubic Fermat curve
Both sm and cm have a period along the real axis of [math]\displaystyle{ \pi_3 = \Beta\bigl( \tfrac13, \tfrac13\bigr) = \tfrac{\sqrt{3}}{2\pi}\Gamma^3\bigl(\tfrac{1}{3}\bigr)\approx 5.29991625 }[/math] with [math]\displaystyle{ \Beta }[/math] the beta function and [math]\displaystyle{ \Gamma }[/math] the gamma function:[5]
- [math]\displaystyle{ \begin{aligned} \tfrac13\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} }[/math]
They satisfy the identity [math]\displaystyle{ \operatorname{cm}^3 z + \operatorname{sm}^3 z = 1 }[/math]. The parametric function [math]\displaystyle{ t \mapsto (\operatorname{cm} t,\, \operatorname{sm} t), }[/math] [math]\displaystyle{ t \in \bigl[{-\tfrac13}\pi_3, \tfrac23\pi_3\bigr] }[/math] parametrizes the cubic Fermat curve [math]\displaystyle{ x^3 + y^3 = 1, }[/math] with [math]\displaystyle{ \tfrac12 t }[/math] representing the signed area lying between the segment from the origin to [math]\displaystyle{ (1,\, 0) }[/math], the segment from the origin to [math]\displaystyle{ (\operatorname{cm} t,\, \operatorname{sm} t) }[/math], 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:
- [math]\displaystyle{ A = \tfrac 12 \int_0^t (x\mathop{dy} -y\mathop{dx}) = \tfrac 12 \int_0^t (\operatorname{cm}^3 t + \operatorname{sm}^3 t)\mathop{dt} = \tfrac 12 \int_0^t dt = \tfrac12 t. }[/math]
Notice that the area between the [math]\displaystyle{ x + y = 0 }[/math] and [math]\displaystyle{ x^3 + y^3 = 1 }[/math] can be broken into three pieces, each of area [math]\displaystyle{ \tfrac16\pi_3 }[/math]:
- [math]\displaystyle{ \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} }[/math]
Symmetries
The function [math]\displaystyle{ \operatorname{sm} z }[/math] has zeros at the complex-valued points [math]\displaystyle{ z = \tfrac1\sqrt{3}\pi_3i(a + b\omega) }[/math] for any integers [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], where [math]\displaystyle{ \omega }[/math] is a cube root of unity, [math]\displaystyle{ \omega = \exp \tfrac23 i \pi = -\tfrac12 + \tfrac\sqrt{3}2i }[/math] (that is, [math]\displaystyle{ a + b\omega }[/math] is an Eisenstein integer). The function [math]\displaystyle{ \operatorname{cm} z }[/math] has zeros at the complex-valued points [math]\displaystyle{ z = \tfrac13\pi_3 + \tfrac1\sqrt{3}\pi_3i(a + b\omega) }[/math]. Both functions have poles at the complex-valued points [math]\displaystyle{ z = -\tfrac13\pi_3 + \tfrac1\sqrt{3}\pi_3i(a + b\omega) }[/math].
On the real line, [math]\displaystyle{ \operatorname{sm}x=0\leftrightarrow x\in\pi_3\mathbb{Z} }[/math], which is analogous to [math]\displaystyle{ \sin x=0\leftrightarrow x\in\pi\mathbb{Z} }[/math].
Fundamental reflections, rotations, and translations
Both cm and sm commute with complex conjugation,
- [math]\displaystyle{ \begin{align} \operatorname{cm} \bar{z} &= \overline{\operatorname{cm} z}, \\ \operatorname{sm} \bar{z} &= \overline{\operatorname{sm} z}. \end{align} }[/math]
Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), the Dixon function cm is invariant under [math]\displaystyle{ \tfrac13 }[/math] turn rotations of the complex plane, and [math]\displaystyle{ \tfrac13 }[/math] turn rotations of the domain of sm cause [math]\displaystyle{ \tfrac13 }[/math] turn rotations of the codomain:
- [math]\displaystyle{ \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} }[/math]
Each Dixon elliptic function is invariant under translations by the Eisenstein integers [math]\displaystyle{ a + b\omega }[/math] scaled by [math]\displaystyle{ \pi_3, }[/math]
- [math]\displaystyle{ \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} }[/math]
Negation of each of cm and sm is equivalent to a [math]\displaystyle{ \tfrac13\pi_3 }[/math] translation of the other,
- [math]\displaystyle{ \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} }[/math]
For [math]\displaystyle{ n \in \mathbb \{0, 1, 2\}, }[/math] translations by [math]\displaystyle{ \tfrac13\pi_3\omega }[/math] give
- [math]\displaystyle{ \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} }[/math]
Specific values
[math]\displaystyle{ z }[/math] | [math]\displaystyle{ \operatorname{cm} z }[/math] | [math]\displaystyle{ \operatorname{sm} z }[/math] |
---|---|---|
[math]\displaystyle{ {-\tfrac13}\pi_3 }[/math] | [math]\displaystyle{ \infty }[/math] | [math]\displaystyle{ \infty }[/math] |
[math]\displaystyle{ {-\tfrac16}\pi_3 }[/math] | [math]\displaystyle{ \sqrt[3]{2} }[/math] | [math]\displaystyle{ -1 }[/math] |
[math]\displaystyle{ 0 }[/math] | [math]\displaystyle{ 1 }[/math] | [math]\displaystyle{ 0 }[/math] |
[math]\displaystyle{ {\tfrac16}\pi_3 }[/math] | [math]\displaystyle{ 1\big/\sqrt[3]{2} }[/math] | [math]\displaystyle{ 1\big/\sqrt[3]{2} }[/math] |
[math]\displaystyle{ {\tfrac13}\pi_3 }[/math] | [math]\displaystyle{ 0 }[/math] | [math]\displaystyle{ 1 }[/math] |
[math]\displaystyle{ {\tfrac12}\pi_3 }[/math] | [math]\displaystyle{ -1 }[/math] | [math]\displaystyle{ \sqrt[3]{2} }[/math] |
[math]\displaystyle{ {\tfrac23}\pi_3 }[/math] | [math]\displaystyle{ \infty }[/math] | [math]\displaystyle{ \infty }[/math] |
More specific values
[math]\displaystyle{ z }[/math] | [math]\displaystyle{ \operatorname{cm} z }[/math] | [math]\displaystyle{ \operatorname{sm} z }[/math] |
---|---|---|
[math]\displaystyle{ {-\tfrac14}\pi_3 }[/math] | [math]\displaystyle{ \frac {1 + \sqrt{3} + \sqrt{2\sqrt{3}}} {2} }[/math] | [math]\displaystyle{ \frac { -1 - \sqrt{3 + 2\sqrt{3}}} { \sqrt[3]{4}} }[/math] |
[math]\displaystyle{ -{\tfrac{1}{12}}\pi_3 }[/math] | [math]\displaystyle{ \frac { -1 + \sqrt{3} + \sqrt{2\sqrt{3}}} {2\sqrt[3]{2}} }[/math] | [math]\displaystyle{ \frac { -1 + \sqrt{3} - \sqrt{2\sqrt{3}}} {2\sqrt[3]{2}} }[/math] |
[math]\displaystyle{ {\tfrac{1}{12}}\pi_3 }[/math] | [math]\displaystyle{ \frac { -1 + \sqrt{3 + 2\sqrt{3}}} { \sqrt[3]{4}} }[/math] | [math]\displaystyle{ \frac {1 + \sqrt{3} - \sqrt{2\sqrt{3}}} {2} }[/math] |
[math]\displaystyle{ {\tfrac14}\pi_3 }[/math] | [math]\displaystyle{ \frac {1 + \sqrt{3} - \sqrt{2\sqrt{3}}} {2} }[/math] | [math]\displaystyle{ \frac { -1 + \sqrt{3 + 2\sqrt{3}}} { \sqrt[3]{4}} }[/math] |
[math]\displaystyle{ {\tfrac{5}{12}}\pi_3 }[/math] | [math]\displaystyle{ \frac { -1 + \sqrt{3} - \sqrt{2\sqrt{3}}} {2\sqrt[3]{2}} }[/math] | [math]\displaystyle{ \frac { -1 + \sqrt{3} + \sqrt{2\sqrt{3}}} {2\sqrt[3]{2}} }[/math] |
[math]\displaystyle{ {\tfrac{7}{12}}\pi_3 }[/math] | [math]\displaystyle{ \frac { -1 - \sqrt{3 + 2\sqrt{3}}} { \sqrt[3]{4}} }[/math] | [math]\displaystyle{ \frac {1 + \sqrt{3} + \sqrt{2\sqrt{3}}} {2} }[/math] |
Sum and difference identities
The Dixon elliptic functions satisfy the argument sum and difference identities:[8]
- [math]\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} }[/math]
These formulas can be used to compute the complex-valued functions in real components:[citation needed]
- [math]\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} }[/math]
Multiple-argument identities
Argument duplication and triplication identities can be derived from the sum identity:[9]
- [math]\displaystyle{ \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} }[/math]
From these formulas it can be deduced that expressions in form [math]\displaystyle{ \operatorname{cm}(\frac {k \pi_3} {2^n3^m}) }[/math] and [math]\displaystyle{ \operatorname{sm}(\frac {k \pi_3} {2^n3^m}) }[/math] are either signless infinities, or origami-constructibles for any [math]\displaystyle{ n, m, k \in \mathbb N }[/math] (In this paragraph, [math]\displaystyle{ \mathbb M = }[/math] set of all origami-constructibles [math]\displaystyle{ \cup \{\infty}\ }[/math]). Because by finding [math]\displaystyle{ \operatorname{cm}(\frac {x} {2}) }[/math], quartic or lesser degree in some cases equation has to be solved as seen from duplication formula which means that if [math]\displaystyle{ \operatorname{cm} x \in \mathbb M }[/math], then [math]\displaystyle{ \operatorname{cm}(\frac {x} {2}) \in \mathbb M }[/math]. To find one-third of argument value of cm, equation which is reductible to cubic or lesser degree in some cases by variable exchange [math]\displaystyle{ t = x^3 }[/math] has to be solved as seen from triplication formula from that follows: if [math]\displaystyle{ \operatorname{cm} x \in \mathbb M }[/math] then [math]\displaystyle{ \operatorname{cm}(\frac {x} {3}) \in \mathbb M }[/math] is true. Statement [math]\displaystyle{ \operatorname{cm} x \in \mathbb M }[/math] [math]\displaystyle{ \Rightarrow }[/math] [math]\displaystyle{ \operatorname{cm}(nx) \in \mathbb M }[/math] is true, because any multiple argument formula is a rational function. If [math]\displaystyle{ \operatorname{cm} x \in \mathbb M }[/math], then [math]\displaystyle{ \operatorname{sm} x \in \mathbb M }[/math] because [math]\displaystyle{ \operatorname{sm} x =\omega^p\,\sqrt[3]{1-\operatorname{cm}^3 x} }[/math] where [math]\displaystyle{ p \in \{0, 1, 2\} }[/math].
Specific value identities
The [math]\displaystyle{ \operatorname{cm} }[/math] function satisfies the identities [math]\displaystyle{ \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} }[/math]
where [math]\displaystyle{ \operatorname{cl} }[/math] is lemniscate cosine and [math]\displaystyle{ \varpi }[/math] is Lemniscate constant.[citation needed]
Power series
The cm and sm functions can be approximated for [math]\displaystyle{ |z| \lt \tfrac13\pi_3 }[/math] by the Taylor series
- [math]\displaystyle{ \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} }[/math]
whose coefficients satisfy the recurrence [math]\displaystyle{ c_0 = s_0 = 1, }[/math][10]
- [math]\displaystyle{ \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} }[/math]
These recurrences result in:[11]
- [math]\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} }[/math]
Relation to other elliptic functions
Weierstrass elliptic function
The equianharmonic Weierstrass elliptic function [math]\displaystyle{ \wp(z) = \wp\bigl(z; 0, \tfrac1{27}\bigr), }[/math] with lattice [math]\displaystyle{ \Lambda = \pi_3\mathbb{Z} \oplus \pi_3\omega\mathbb{Z} }[/math] a scaling of the Eisenstein integers, can be defined as:[12]
- [math]\displaystyle{ \wp(z) = \frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\!\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right) }[/math]
The function [math]\displaystyle{ \wp(z) }[/math] solves the differential equation:
- [math]\displaystyle{ \wp'(z)^2 = 4\wp(z)^3 - \tfrac1{27} }[/math]
We can also write it as the inverse of the integral:
- [math]\displaystyle{ z = \int_{\infty}^{\wp(z)} \frac{dw}{\sqrt{4w^3 - \tfrac1{27}}} }[/math]
In terms of [math]\displaystyle{ \wp(z) }[/math], the Dixon elliptic functions can be written:[13]
- [math]\displaystyle{ \operatorname{cm} z = \frac{3\wp'(z) + 1}{3\wp'(z) - 1},\ \operatorname{sm} z = \frac{-6\wp(z)}{3\wp'(z) - 1} }[/math]
Likewise, the Weierstrass elliptic function [math]\displaystyle{ \wp(z) = \wp\bigl(z; 0, \tfrac1{27}\bigr) }[/math] can be written in terms of Dixon elliptic functions:
- [math]\displaystyle{ \wp'(z) = \frac{\operatorname{cm} z + 1}{3(\operatorname{cm} z - 1)},\ \wp(z) = \frac{-\operatorname{sm} z}{3(\operatorname{cm} z - 1)} }[/math]
Jacobi elliptic functions
The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley.[14] Let [math]\displaystyle{ k = e^{5 i \pi / 6} }[/math], [math]\displaystyle{ \theta = 3^{\frac{1}{4}} e^{5 i \pi / 12} }[/math], [math]\displaystyle{ s = \operatorname {sn}(u,k) }[/math], [math]\displaystyle{ c = \operatorname {cn}(u,k) }[/math], and [math]\displaystyle{ d = \operatorname {dn}(u,k) }[/math]. Then, let
- [math]\displaystyle{ \xi(u) = \frac{-1 + \theta scd}{1 + \theta scd} }[/math], [math]\displaystyle{ \eta(u) = \frac{2^{1/3} \left( 1+ \theta^2 s^2\right)}{1 + \theta scd} }[/math].
Finally, the Dixon elliptic functions are as so:
- [math]\displaystyle{ \operatorname {sm}(z) = \xi \left(\frac{z + \pi_3/6}{2^{1/3} \theta}\right) }[/math], [math]\displaystyle{ \operatorname {cm}(z) = \eta \left(\frac{z + \pi_3/6}{2^{1/3} \theta}\right) }[/math].
Generalized trigonometry
Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an [math]\displaystyle{ n = 2 }[/math] case, and the functions sm and cm as an [math]\displaystyle{ n = 3 }[/math] case.[15]
For example, defining [math]\displaystyle{ \pi_n = \Beta\bigl(\tfrac1n, \tfrac1n\bigr) }[/math] and [math]\displaystyle{ \sin_n z,\,\cos_n z }[/math] the inverses of an integral:
- [math]\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}} }[/math]
The area in the positive quadrant under the curve [math]\displaystyle{ x^n + y^n = 1 }[/math] is
- [math]\displaystyle{ \int_0^{1} (1 - x^n)^{1/n}\mathop{dx} = \frac{\pi_n}{2n} }[/math].
The quartic [math]\displaystyle{ n = 4 }[/math] 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.[16]
See also
- Eisenstein integer
- Elliptic function
- Lee conformal world in a tetrahedron
- Schwarz–Christoffel mapping
Notes
- ↑ Dixon (1890), Dillner (1873). Dillner uses the symbols [math]\displaystyle{ W = \operatorname{sm},\ W_1 = \operatorname{cm}. }[/math]
- ↑ Dixon (1890), Van Fossen Conrad & Flajolet (2005), Robinson (2019).
- ↑ The mapping for a general regular polygon is described in Schwarz (1869).
- ↑ van Fossen Conrad & Flajolet (2005) p. 6.
- ↑ Dillner (1873) calls the period [math]\displaystyle{ 3w }[/math]. Dixon (1890) calls it [math]\displaystyle{ 3\lambda }[/math]; Adams (1925) and Robinson (2019) each call it [math]\displaystyle{ 3K }[/math]. Van Fossen Conrad & Flajolet (2005) call it [math]\displaystyle{ \pi_3 }[/math]. Also see OEIS A197374.
- ↑ Dixon (1890), Van Fossen Conrad & Flajolet (2005)
- ↑ Dark areas represent zeros, and bright areas represent poles. As the argument of [math]\displaystyle{ \operatorname{sm}z }[/math] goes from [math]\displaystyle{ -\pi }[/math] to [math]\displaystyle{ \pi }[/math], the colors go through cyan, blue ([math]\displaystyle{ \operatorname{Arg}\approx -\pi/2 }[/math]), magneta, red ([math]\displaystyle{ \operatorname{Arg}\approx 0 }[/math]), orange, yellow ([math]\displaystyle{ \operatorname{Arg}\approx\pi/2 }[/math]), green, and back to cyan ([math]\displaystyle{ \operatorname{Arg}\approx\pi }[/math]).
- ↑ Dixon (1890), Adams (1925)
- ↑ Dixon (1890), p. 185–186. Robinson (2019).
- ↑ Adams (1925)
- ↑ van Fossen Conrad & Flajolet (2005). Also see OEIS A104133, A104134.
- ↑ Reinhardt & Walker (2010)
- ↑ Chapling (2018), Robinson (2019). Adams (1925) instead expresses the Dixon elliptic functions in terms of the Weierstrass elliptic function [math]\displaystyle{ \wp(z; 0, -1). }[/math]
- ↑ van Fossen Conrad & Flajolet (2005), p.38
- ↑ Lundberg (1879), Grammel (1948), Shelupsky (1959), Burgoyne (1964), Gambini, Nicoletti, & Ritelli (2021).
- ↑ 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 [math]\displaystyle{ \int dx / (1 - x^3){}^{2/3} }[/math] 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 x3 + y3 − 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 x3 + y3 − 3αxy = 1". Quarterly Journal of Pure and Applied Mathematics XXIV: 167–233. https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify={%22pages%22:%5b179%5d}.
- A. Dixon (1894) The elementary properties of the elliptic functions. MacMillian. https://archive.org/details/elempropellipt00dixorich/
- Van Fossen Conrad, Eric (2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion". Séminaire Lotharingien de Combinatoire 54: Art. B54g, 44. Bibcode: 2005math......7268V.
- 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
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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 [math]\displaystyle{ x^3+y^3=z^3 }[/math], 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/
Original source: https://en.wikipedia.org/wiki/Dixon elliptic functions.
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