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Dixon elliptic functions
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<h2 id="definition">Definition</h2> <p>The functions sm and cm can be defined as the solutions to the <a href="/facts/Initial_value_problem/PqmJfGML">initial value problem</a>:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> d d z cm z = − sm 2 z , d d z sm z = cm 2 z , cm ( 0 ) = 1 , sm ( 0 ) = 0 {\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} <p>Or as the inverse of the <a href="/facts/Schwarz%25E2%2580%2593Christoffel_mapping/vzHbRDDG">Schwarz–Christoffel mapping</a> from the complex unit disk to an equilateral triangle, the <a href="/facts/Abelian_integral/2oLFBNWk">Abelian integral</a>:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> z = ∫ 0 sm z d w ( 1 − w 3 ) 2 / 3 = ∫ cm z 1 d w ( 1 − w 3 ) 2 / 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}}}} <p>which can also be expressed using the <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric function</a>:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> sm − 1 ( z ) = z 2 F 1 ( 1 3 , 2 3 ; 4 3 ; z 3 ) {\displaystyle \operatorname {sm} ^{-1}(z)=z\;{}_{2}F_{1}{\bigl (}{\tfrac {1}{3}},{\tfrac {2}{3}};{\tfrac {4}{3}};z^{3}{\bigr )}} <h2 id="parametrization-of-the-cubic-fermat-curve">Parametrization of the cubic Fermat curve</h2> <p>Both sm and cm have a period along the real axis of π 3 = B ( 1 3 , 1 3 ) = 3 2 π Γ 3 ( 1 3 ) ≈ 5.29991625 {\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 B {\displaystyle \mathrm {B} } the <a href="/facts/Beta_function/Dcm1JAxb">beta function</a> and Γ {\displaystyle \Gamma } the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> 1 3 π 3 = ∫ − ∞ 0 d x ( 1 − x 3 ) 2 / 3 = ∫ 0 1 d x ( 1 − x 3 ) 2 / 3 = ∫ 1 ∞ d x ( 1 − x 3 ) 2 / 3 ≈ 1.76663875 {\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}}} <p>They satisfy the identity cm 3 z + sm 3 z = 1 {\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1} . The parametric function t ↦ ( cm t , sm t ) , {\displaystyle t\mapsto (\operatorname {cm} t,\,\operatorname {sm} t),} t ∈ [ − 1 3 π 3 , 2 3 π 3 ] {\displaystyle t\in {\bigl [}{-{\tfrac {1}{3}}}\pi _{3},{\tfrac {2}{3}}\pi _{3}{\bigr ]}} parametrizes the cubic <a href="/facts/Fermat_curve/31LFyJ5p">Fermat curve</a> x 3 + y 3 = 1 , {\displaystyle x^{3}+y^{3}=1,} with 1 2 t {\displaystyle {\tfrac {1}{2}}t} representing the signed area lying between the segment from the origin to ( 1 , 0 ) {\displaystyle (1,\,0)} , the segment from the origin to ( cm t , sm t ) {\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.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> To see why, apply <a href="/facts/Green%2527s_theorem/WBDk654A">Green's theorem</a>: </p> A = 1 2 ∫ 0 t ( x d y − y d x ) = 1 2 ∫ 0 t ( cm 3 t + sm 3 t ) d t = 1 2 ∫ 0 t d t = 1 2 t . {\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.} <p>Notice that the area between the x + y = 0 {\displaystyle x+y=0} and x 3 + y 3 = 1 {\displaystyle x^{3}+y^{3}=1} can be broken into three pieces, each of area 1 6 π 3 {\displaystyle {\tfrac {1}{6}}\pi _{3}} : </p> 1 2 π 3 = ∫ − ∞ ∞ ( ( 1 − x 3 ) 1 / 3 + x ) d x 1 6 π 3 = ∫ − ∞ 0 ( ( 1 − x 3 ) 1 / 3 + x ) d x = ∫ 0 1 ( 1 − x 3 ) 1 / 3 d x . {\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}}} <h2 id="symmetries">Symmetries</h2> <p>The function sm z {\displaystyle \operatorname {sm} z} has <a href="/facts/Zeros_and_poles/gnA3U8VP">zeros</a> at the complex-valued points z = 1 3 π 3 i ( a + b ω ) {\displaystyle z={\tfrac {1}{\sqrt {3}}}\pi _{3}i(a+b\omega )} for any integers a {\displaystyle a} and b {\displaystyle b} , where ω {\displaystyle \omega } is a cube <a href="/facts/Root_of_unity/EqH0ho1Y">root of unity</a>, ω = exp 2 3 i π = − 1 2 + 3 2 i {\displaystyle \omega =\exp {\tfrac {2}{3}}i\pi =-{\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i} (that is, a + b ω {\displaystyle a+b\omega } is an <a href="/facts/Eisenstein_integer/UGQmFI8c">Eisenstein integer</a>). The function cm z {\displaystyle \operatorname {cm} z} has zeros at the complex-valued points z = 1 3 π 3 + 1 3 π 3 i ( a + b ω ) {\displaystyle z={\tfrac {1}{3}}\pi _{3}+{\tfrac {1}{\sqrt {3}}}\pi _{3}i(a+b\omega )} . Both functions have <a href="/facts/Zeros_and_poles/gnA3U8VP">poles</a> at the complex-valued points z = − 1 3 π 3 + 1 3 π 3 i ( a + b ω ) {\displaystyle z=-{\tfrac {1}{3}}\pi _{3}+{\tfrac {1}{\sqrt {3}}}\pi _{3}i(a+b\omega )} . </p><p>On the real line, sm x = 0 ↔ x ∈ π 3 Z {\displaystyle \operatorname {sm} x=0\leftrightarrow x\in \pi _{3}\mathbb {Z} } , which is analogous to sin x = 0 ↔ x ∈ π Z {\displaystyle \sin x=0\leftrightarrow x\in \pi \mathbb {Z} } . </p> <h3>Fundamental reflections, rotations, and translations</h3> <p>Both cm and sm commute with complex conjugation, </p> cm z ¯ = cm z ¯ , sm z ¯ = sm z ¯ . {\displaystyle {\begin{aligned}\operatorname {cm} {\bar {z}}&={\overline {\operatorname {cm} z}},\\\operatorname {sm} {\bar {z}}&={\overline {\operatorname {sm} z}}.\end{aligned}}} <p>Analogous to the parity of trigonometric functions (cosine an <a href="/facts/Even_function/tGHWJqAa">even function</a> and sine an <a href="/facts/Odd_function/tGHWJqAa">odd function</a>), the Dixon function cm is invariant under 1 3 {\textstyle {\tfrac {1}{3}}} turn rotations of the complex plane, and 1 3 {\textstyle {\tfrac {1}{3}}} turn rotations of the domain of sm cause 1 3 {\displaystyle {\tfrac {1}{3}}} turn rotations of the codomain: </p> cm ω z = cm z = cm ω 2 z , sm ω z = ω sm z = ω 2 sm ω 2 z . {\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}}} <p>Each Dixon elliptic function is invariant under translations by the Eisenstein integers a + b ω {\displaystyle a+b\omega } scaled by π 3 , {\displaystyle \pi _{3},} </p> cm ( z + π 3 ( a + b ω ) ) = cm z , sm ( z + π 3 ( a + b ω ) ) = sm z . {\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}}} <p>Negation of each of cm and sm is equivalent to a 1 3 π 3 {\displaystyle {\tfrac {1}{3}}\pi _{3}} translation of the other, </p> cm ( − z ) = 1 cm z = sm ( z + 1 3 π 3 ) , sm ( − z ) = − sm z cm z = 1 sm ( z − 1 3 π 3 ) = cm ( z + 1 3 π 3 ) . {\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}}} <p>For n ∈ { 0 , 1 , 2 } , {\displaystyle n\in \mathbb {\{} 0,1,2\},} translations by 1 3 π 3 ω {\displaystyle {\tfrac {1}{3}}\pi _{3}\omega } give </p> cm ( z + 1 3 ω n π 3 ) = ω 2 n − sm z cm z , sm ( z + 1 3 ω n π 3 ) = ω n 1 cm z . {\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}}} <h3>Specific values</h3> <table><tbody><tr><th> z {\displaystyle z} </th><th> cm z {\displaystyle \operatorname {cm} z} </th><th> sm z {\displaystyle \operatorname {sm} z} </th></tr><tr><td> − 1 3 π 3 {\displaystyle {-{\tfrac {1}{3}}}\pi _{3}} </td><td> ∞ {\displaystyle \infty } </td><td> ∞ {\displaystyle \infty } </td></tr><tr><td> − 1 6 π 3 {\displaystyle {-{\tfrac {1}{6}}}\pi _{3}} </td><td> 2 3 {\displaystyle {\sqrt[{3}]{2}}} </td><td> − 1 {\displaystyle -1} </td></tr><tr><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td><td> 0 {\displaystyle 0} </td></tr><tr><td> 1 6 π 3 {\displaystyle {\tfrac {1}{6}}\pi _{3}} </td><td> 1 / 2 3 {\displaystyle 1{\big /}{\sqrt[{3}]{2}}} </td><td> 1 / 2 3 {\displaystyle 1{\big /}{\sqrt[{3}]{2}}} </td></tr><tr><td> 1 3 π 3 {\displaystyle {\tfrac {1}{3}}\pi _{3}} </td><td> 0 {\displaystyle 0} </td><td> 1 {\displaystyle 1} </td></tr><tr><td> 1 2 π 3 {\displaystyle {\tfrac {1}{2}}\pi _{3}} </td><td> − 1 {\displaystyle -1} </td><td> 2 3 {\displaystyle {\sqrt[{3}]{2}}} </td></tr><tr><td> 2 3 π 3 {\displaystyle {\tfrac {2}{3}}\pi _{3}} </td><td> ∞ {\displaystyle \infty } </td><td> ∞ {\displaystyle \infty } </td></tr></tbody></table> <h4>More specific values</h4> <table><tbody><tr><th> z {\displaystyle z} </th><th> cm z {\displaystyle \operatorname {cm} z} </th><th> sm z {\displaystyle \operatorname {sm} z} </th></tr><tr><td> − 1 4 π 3 {\displaystyle {-{\tfrac {1}{4}}}\pi _{3}} </td><td> 1 + 3 + 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}} </td><td> − 1 − 3 + 2 3 4 3 {\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} </td></tr><tr><td> − 2 9 π 3 {\displaystyle -{\tfrac {2}{9}}\pi _{3}} </td><td> 3 6 2 sin ( 1 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}} </td><td> − 2 cos ( 1 18 π ) 3 6 {\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt[{6}]{3}}}} </td></tr><tr><td> − 1 9 π 3 {\displaystyle -{\tfrac {1}{9}}\pi _{3}} </td><td> 2 sin ( 2 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt[{6}]{3}}}} </td><td> − 3 6 2 cos ( 1 18 π ) {\displaystyle -{\frac {\sqrt[{6}]{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}} </td></tr><tr><td> − 1 12 π 3 {\displaystyle -{\tfrac {1}{12}}\pi _{3}} </td><td> − 1 + 3 + 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} </td><td> − 1 + 3 − 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} </td></tr><tr><td> 1 12 π 3 {\displaystyle {\tfrac {1}{12}}\pi _{3}} </td><td> − 1 + 3 + 2 3 4 3 {\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} </td><td> 1 + 3 − 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}} </td></tr><tr><td> 1 9 π 3 {\displaystyle {\tfrac {1}{9}}\pi _{3}} </td><td> 3 6 2 sin ( 2 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}} </td><td> 2 sin ( 1 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt[{6}]{3}}}} </td></tr><tr><td> 2 9 π 3 {\displaystyle {\tfrac {2}{9}}\pi _{3}} </td><td> 2 sin ( 1 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt[{6}]{3}}}} </td><td> 3 6 2 sin ( 2 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}} </td></tr><tr><td> 1 4 π 3 {\displaystyle {\tfrac {1}{4}}\pi _{3}} </td><td> 1 + 3 − 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}} </td><td> − 1 + 3 + 2 3 4 3 {\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} </td></tr><tr><td> 5 12 π 3 {\displaystyle {\tfrac {5}{12}}\pi _{3}} </td><td> − 1 + 3 − 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} </td><td> − 1 + 3 + 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} </td></tr><tr><td> 4 9 π 3 {\displaystyle {\tfrac {4}{9}}\pi _{3}} </td><td> − 3 6 2 cos ( 1 18 π ) {\displaystyle -{\frac {\sqrt[{6}]{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}} </td><td> 2 sin ( 2 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt[{6}]{3}}}} </td></tr><tr><td> 5 9 π 3 {\displaystyle {\tfrac {5}{9}}\pi _{3}} </td><td> − 2 cos ( 1 18 π ) 3 6 {\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt[{6}]{3}}}} </td><td> 3 6 2 sin ( 1 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}} </td></tr><tr><td> 7 12 π 3 {\displaystyle {\tfrac {7}{12}}\pi _{3}} </td><td> − 1 − 3 + 2 3 4 3 {\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} </td><td> 1 + 3 + 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}} </td></tr></tbody></table> <h2 id="sum-and-difference-identities">Sum and difference identities</h2> <p>The Dixon elliptic functions satisfy the argument sum and difference identities:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> cm ( u + v ) = sm u cm u − sm v cm v sm u cm 2 v − cm 2 u sm v cm ( u − v ) = cm 2 u cm v − sm u sm 2 v cm u cm 2 v − sm 2 u sm v sm ( u + v ) = sm 2 u cm v − cm u sm 2 v sm u cm 2 v − cm 2 u sm v sm ( u − v ) = sm u cm u − sm v cm v cm u cm 2 v − sm 2 u sm v {\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}}} <p>These formulas can be used to compute the complex-valued functions in real components: </p> cm ( x + ω y ) = sm x cm x − ω sm y cm y sm x cm 2 y − ω cm 2 x sm y = cm x ( sm 2 x cm 2 y + cm x sm 2 y cm y + sm x cm 2 x sm y ) sm 2 x cm 4 y + sm x cm 2 x sm y cm 2 y + cm 4 x sm 2 y + ω sm x sm y ( cm 3 x − cm 3 y ) sm 2 x cm 4 y + sm x cm 2 x sm y cm 2 y + cm 4 x sm 2 y sm ( x + ω y ) = sm 2 x cm y − ω 2 cm x sm 2 y sm x cm 2 y − ω cm 2 x sm y = sm x ( sm x cm x cm 2 y + sm y cm 3 x + sm y cm 3 y ) sm 2 x cm 4 y + sm x cm 2 x sm y cm 2 y + cm 4 x sm 2 y + ω sm y ( sm x cm 3 x + sm x cm 3 y + cm 2 x sm y cm y ) sm 2 x cm 4 y + sm x cm 2 x sm y cm 2 y + cm 4 x sm 2 y {\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}}} <h2 id="multiple-argument-identities">Multiple-argument identities</h2> <p>Argument duplication and triplication identities can be derived from the sum identity:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> cm 2 u = cm 3 u − sm 3 u cm u ( 1 + sm 3 u ) = 2 cm 3 u − 1 2 cm u − cm 4 u , sm 2 u = sm u ( 1 + cm 3 u ) cm u ( 1 + sm 3 u ) = 2 sm u − sm 4 u 2 cm u − cm 4 u , cm 3 u = cm 9 u − 6 cm 6 u + 3 cm 3 u + 1 cm 9 u + 3 cm 6 u − 6 cm 3 u + 1 , sm 3 u = 3 sm u cm u ( sm 3 u cm 3 u − 1 ) cm 9 u + 3 cm 6 u − 6 cm 3 u + 1 . {\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}}} <h2 id="specific-value-identities">Specific value identities</h2> <p>The cm {\displaystyle \operatorname {cm} } function satisfies the identities cm 2 9 π 3 = − cm 1 9 π 3 cm 4 9 π 3 , cm 1 4 π 3 = cl 1 3 ϖ , {\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}}} </p><p>where cl {\displaystyle \operatorname {cl} } is <a href="/facts/Lemniscate_elliptic_functions/cHWb3moy">lemniscate cosine</a> and ϖ {\displaystyle \varpi } is <a href="/facts/Lemniscate_constant/jzkcBdBL">Lemniscate constant</a>. </p> <h2 id="power-series">Power series</h2> <p>The cm and sm functions can be approximated for | z | < 1 3 π 3 {\displaystyle |z|<{\tfrac {1}{3}}\pi _{3}} by the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> </p> cm z = c 0 + c 1 z 3 + c 2 z 6 + c 3 z 9 + ⋯ + c n z 3 n + ⋯ sm z = s 0 z + s 1 z 4 + s 2 z 7 + s 3 z 10 + ⋯ + s n z 3 n + 1 + ⋯ {\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}}} <p>whose coefficients satisfy the recurrence c 0 = s 0 = 1 , {\displaystyle c_{0}=s_{0}=1,} <a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> c n = − 1 3 n ∑ k = 0 n − 1 s k s n − 1 − k s n = 1 3 n + 1 ∑ k = 0 n c k c n − k {\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}}} <p>These recurrences result in:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> cm z = 1 − 1 3 z 3 + 1 18 z 6 − 23 2268 z 9 + 25 13608 z 12 − 619 1857492 z 15 + ⋯ sm z = z − 1 6 z 4 + 2 63 z 7 − 13 2268 z 10 + 23 22113 z 13 − 2803 14859936 z 16 + ⋯ {\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}}} <h2 id="relation-to-other-elliptic-functions">Relation to other elliptic functions</h2> <h3>Weierstrass elliptic function</h3> <p>The <a href="/facts/Equianharmonic/fgU3XnRs">equianharmonic</a> <a href="/facts/Weierstrass_elliptic_functions/GHRXDknk">Weierstrass elliptic function</a> ℘ ( z ) = ℘ ( z ; 0 , 1 27 ) , {\displaystyle \wp (z)=\wp {\bigl (}z;0,{\tfrac {1}{27}}{\bigr )},} with <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> Λ = π 3 Z ⊕ π 3 ω Z {\displaystyle \Lambda =\pi _{3}\mathbb {Z} \oplus \pi _{3}\omega \mathbb {Z} } a scaling of the Eisenstein integers, can be defined as:<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> ℘ ( z ) = 1 z 2 + ∑ λ ∈ Λ ∖ { 0 } ( 1 ( z − λ ) 2 − 1 λ 2 ) {\displaystyle \wp (z)={\frac {1}{z^{2}}}+\sum _{\lambda \in \Lambda \smallsetminus \{0\}}\!\left({\frac {1}{(z-\lambda )^{2}}}-{\frac {1}{\lambda ^{2}}}\right)} <p>The function ℘ ( z ) {\displaystyle \wp (z)} solves the differential equation: </p> ℘ ′ ( z ) 2 = 4 ℘ ( z ) 3 − 1 27 {\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-{\tfrac {1}{27}}} <p>We can also write it as the inverse of the integral: </p> z = ∫ ∞ ℘ ( z ) d w 4 w 3 − 1 27 {\displaystyle z=\int _{\infty }^{\wp (z)}{\frac {dw}{\sqrt {4w^{3}-{\tfrac {1}{27}}}}}} <p>In terms of ℘ ( z ) {\displaystyle \wp (z)} , the Dixon elliptic functions can be written:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> cm z = 3 ℘ ′ ( z ) + 1 3 ℘ ′ ( z ) − 1 , sm z = − 6 ℘ ( z ) 3 ℘ ′ ( z ) − 1 {\displaystyle \operatorname {cm} z={\frac {3\wp '(z)+1}{3\wp '(z)-1}},\ \operatorname {sm} z={\frac {-6\wp (z)}{3\wp '(z)-1}}} <p>Likewise, the Weierstrass elliptic function ℘ ( z ) = ℘ ( z ; 0 , 1 27 ) {\displaystyle \wp (z)=\wp {\bigl (}z;0,{\tfrac {1}{27}}{\bigr )}} can be written in terms of Dixon elliptic functions: </p> ℘ ′ ( z ) = cm z + 1 3 ( cm z − 1 ) , ℘ ( z ) = − sm z 3 ( cm z − 1 ) {\displaystyle \wp '(z)={\frac {\operatorname {cm} z+1}{3(\operatorname {cm} z-1)}},\ \wp (z)={\frac {-\operatorname {sm} z}{3(\operatorname {cm} z-1)}}} <h3>Jacobi elliptic functions</h3> <p>The Dixon elliptic functions can also be expressed using <a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic functions</a>, which was first observed by <a href="/facts/Arthur_Cayley/NBd7QkeE">Cayley</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> Let k = e 5 i π / 6 {\displaystyle k=e^{5i\pi /6}} , θ = 3 1 4 e 5 i π / 12 {\displaystyle \theta =3^{\frac {1}{4}}e^{5i\pi /12}} , s = sn ( u , k ) {\displaystyle s=\operatorname {sn} (u,k)} , c = cn ( u , k ) {\displaystyle c=\operatorname {cn} (u,k)} , and d = dn ( u , k ) {\displaystyle d=\operatorname {dn} (u,k)} . Then, let </p> ξ ( u ) = − 1 + θ s c d 1 + θ s c d {\displaystyle \xi (u)={\frac {-1+\theta scd}{1+\theta scd}}} , η ( u ) = 2 1 / 3 ( 1 + θ 2 s 2 ) 1 + θ s c d . {\displaystyle \eta (u)={\frac {2^{1/3}\left(1+\theta ^{2}s^{2}\right)}{1+\theta scd}}.} <p>Finally, the Dixon elliptic functions are as so: </p> sm ( z ) = ξ ( z + π 3 / 6 2 1 / 3 θ ) , {\displaystyle \operatorname {sm} (z)=\xi \left({\frac {z+\pi _{3}/6}{2^{1/3}\theta }}\right),} cm ( z ) = η ( z + π 3 / 6 2 1 / 3 θ ) . {\displaystyle \operatorname {cm} (z)=\eta \left({\frac {z+\pi _{3}/6}{2^{1/3}\theta }}\right).} <h2 id="generalized-trigonometry">Generalized trigonometry</h2> <p>Several definitions of generalized trigonometric functions include the usual trigonometric sine and cosine as an n = 2 {\displaystyle n=2} case, and the functions sm and cm as an n = 3 {\displaystyle n=3} case.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>For example, defining π n = B ( 1 n , 1 n ) {\displaystyle \pi _{n}=\mathrm {B} {\bigl (}{\tfrac {1}{n}},{\tfrac {1}{n}}{\bigr )}} and sin n z , cos n z {\displaystyle \sin _{n}z,\,\cos _{n}z} the inverses of an integral: </p> z = ∫ 0 sin n z d w ( 1 − w n ) ( n − 1 ) / n = ∫ cos n z 1 d w ( 1 − w n ) ( n − 1 ) / n {\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}}}} <p>The area in the positive quadrant under the curve x n + y n = 1 {\displaystyle x^{n}+y^{n}=1} is </p> ∫ 0 1 ( 1 − x n ) 1 / n d x = π n 2 n . {\displaystyle \int _{0}^{1}(1-x^{n})^{1/n}\,\mathrm {d} x={\frac {\pi _{n}}{2n}}.} <p>The quartic n = 4 {\displaystyle n=4} case results in a square lattice in the complex plane, related to the <a href="/facts/Lemniscate_elliptic_functions/cHWb3moy">lemniscate elliptic functions</a>. </p> <h2 id="applications">Applications</h2> <p>The Dixon elliptic functions are <a href="/facts/Conformal_map/cEpQnMad">conformal maps</a> from an equilateral triangle to a disk, and are therefore helpful for constructing polyhedral <a href="/facts/Conformal_map_projection/ExV5g5AL">conformal map projections</a> involving equilateral triangles, for example projecting the sphere onto a triangle, hexagon, <a href="/facts/Lee_conformal_world_in_a_tetrahedron/6z8W8utd">tetrahedron</a>, octahedron, or icosahedron.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Eisenstein_integer/UGQmFI8c">Eisenstein integer</a></li> <li><a href="/facts/Elliptic_function/mF03oI04">Elliptic function</a> <ul><li><a href="/facts/Abel_elliptic_functions/JqolraWM">Abel elliptic functions</a></li> <li><a href="/facts/Jacobi_elliptic_functions/BLk6Rxew">Jacobi elliptic functions</a></li> <li><a href="/facts/Lemniscate_elliptic_functions/cHWb3moy">Lemniscate elliptic functions</a></li> <li><a href="/facts/Weierstrass_elliptic_function/GHRXDknk">Weierstrass elliptic function</a></li></ul></li> <li><a href="/facts/Lee_conformal_world_in_a_tetrahedron/6z8W8utd">Lee conformal world in a tetrahedron</a></li> <li><a href="/facts/Schwarz%25E2%2580%2593Christoffel_mapping/vzHbRDDG">Schwarz–Christoffel mapping</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Oscar_S._Adams/1ss0vLlI">O. S. Adams</a> (1925). <i>Elliptic functions applied to conformal world maps</i> (No. 297). US Government Printing Office. <a href="ftp://ftp.library.noaa.gov/docs.lib/htdocs/rescue/cgs_specpubs/QB275U35no1121925.pdf">ftp://ftp.library.noaa.gov/docs.lib/htdocs/rescue/cgs_specpubs/QB275U35no1121925.pdf</a></li> <li>R. Bacher & P. Flajolet (2010) “Pseudo-factorials, elliptic functions, and continued fractions” <i>The Ramanujan journal</i> 21(1), 71–97. <a href="https://arxiv.org/pdf/0901.1379.pdf">https://arxiv.org/pdf/0901.1379.pdf</a></li> <li><a href="/facts/Arthur_Cayley/NBd7QkeE">A. Cayley</a> (1882) “Reduction of ∫ d x / ( 1 − x 3 ) 2 / 3 {\textstyle \int dx/(1-x^{3}){}^{2/3}} to elliptic integrals”. <i>Messenger of Mathematics</i> 11, 142–143. <a href="https://gdz.sub.uni-goettingen.de/id/PPN599484047_0011?tify={%22pages%22:%5b146%5d}">https://gdz.sub.uni-goettingen.de/id/PPN599484047_0011?tify={%22pages%22:%5b146%5d}</a></li> <li>F. D. Burgoyne (1964) “Generalized trigonometric functions”. <i>Mathematics of Computation</i> 18(86), 314–316. <a href="https://www.jstor.org/stable/2003310">https://www.jstor.org/stable/2003310</a></li> <li>A. Cayley (1883) “On the elliptic function solution of the equation <i>x</i>3 + <i>y</i>3 − 1 = 0”, <i>Proceedings of the Cambridge Philosophical Society</i> 4, 106–109. <a href="https://archive.org/details/proceedingsofcam4188083camb/page/106/">https://archive.org/details/proceedingsofcam4188083camb/page/106/</a></li> <li>R. Chapling (2016) “Invariant Meromorphic Functions on the Wallpaper Groups”. <a href="https://arxiv.org/pdf/1608.05677">https://arxiv.org/pdf/1608.05677</a></li> <li>J. F. Cox (1935) “Répresentation de la surface entière de la terre dans une triangle équilatéral”, <i>Bulletin de la Classe des Sciences, Académie Royale de Belgique 5e</i>, 21, 66–71.</li> <li>G. Dillner (1873) “Traité de calcul géométrique supérieur”, Chapter 16, <i>Nova acta Regiae Societatis Scientiarum Upsaliensis, Ser. III</i> 8, 94–102. <a href="https://archive.org/details/novaactaregiaeso38kung/page/94/">https://archive.org/details/novaactaregiaeso38kung/page/94/</a></li> <li><a href="/facts/Alfred_Cardew_Dixon/vkhTloGh">Dixon, A. C.</a> (1890). <a href="https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify={%22pages%22:%5b179%5d}">"On the doubly periodic functions arising out of the curve <i>x</i>3 + <i>y</i>3 − 3<i>αxy</i> = 1"</a>. <i>Quarterly Journal of Pure and Applied Mathematics</i>. XXIV: 167–233.</li> <li>A. Dixon (1894) <i>The elementary properties of the elliptic functions</i>. MacMillian. <a href="https://archive.org/details/elempropellipt00dixorich/">https://archive.org/details/elempropellipt00dixorich/</a></li> <li>Van Fossen Conrad, Eric; <a href="/facts/Philippe_Flajolet/8C9NqUlD">Flajolet, Philippe</a> (2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion". <i><a href="/facts/S%25C3%25A9minaire_Lotharingien_de_Combinatoire/sRFE1Jnl">Séminaire Lotharingien de Combinatoire</a></i>. 54: Art. B54g, 44. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0507268">math/0507268</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005math......7268V">2005math......7268V</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2223029">2223029</a>.</li> <li>A. Gambini, G. Nicoletti, & D. Ritelli (2021) “Keplerian trigonometry”. <i>Monatshefte für Mathematik</i> 195(1), 55–72. <a href="https://doi.org/10.1007/s00605-021-01512-0">https://doi.org/10.1007/s00605-021-01512-0</a></li> <li>R. Grammel (1948) “Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen”. <i>Archiv der Mathematik</i> 1(1), 47–51. <a href="https://doi.org/10.1007/BF02038206">https://doi.org/10.1007/BF02038206</a></li> <li>J. C. Langer & D. A. Singer (2014) “The Trefoil”. <i>Milan Journal of Mathematics</i> 82(1), 161–182. <a href="https://case.edu/artsci/math/langer/jlpreprints/Trefoil.pdf">https://case.edu/artsci/math/langer/jlpreprints/Trefoil.pdf</a></li> <li>M. Laurent (1949) “Tables de la fonction elliptique de Dixon pour l’intervalle 0-0, 1030”. <i>Bulletin de l’Académie Royale des Sciences de Belgique Classe des Sciences</i>, 35, 439–450.</li> <li><a href="/facts/L._P._Lee/8ffsr92N">L. P. Lee</a> (1973) “The Conformal Tetrahedric Projection with some Practical Applications”. <i>The Cartographic Journal</i>, 10(1), 22–28. <a href="https://doi.org/10.1179/caj.1973.10.1.22">https://doi.org/10.1179/caj.1973.10.1.22</a></li> <li><a href="/facts/L._P._Lee/8ffsr92N">L. P. Lee</a> (1976) <a href="https://archive.org/details/conformalproject0000leel"><i>Conformal Projections Based on Elliptic Functions</i></a>. Toronto: B. V. Gutsell, York University. <i>Cartographica Monographs</i> No. 16. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-919870-16-3. Supplement No. 1 to <a href="https://www.utpjournals.press/toc/cart/13/1"><i>The Canadian Cartographer</i> 13</a>.</li> <li>E. Lundberg (1879) “Om hypergoniometriska funktioner af komplexa variabla”. Manuscript, 1879. Translation by Jaak Peetre “On hypergoniometric functions of complex variables”. <a href="https://web.archive.org/web/20161024183030/http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps">https://web.archive.org/web/20161024183030/http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps</a></li> <li>J. Magis (1938) “Calcul du canevas de la représentation conforme de la sphère entière dans un triangle équilatéral”. <i>Bulletin Géodésique</i> 59(1), 247–256. <a href="https://doi.org/10.1007/BF03029866">http://doi.org/10.1007/BF03029866</a></li> <li>M. D. McIlroy (2011) “Wallpaper maps”. <i>Dependable and Historic Computing</i>. Springer. 358–375. <a href="https://link.springer.com/chapter/10.1007/978-3-642-24541-1_27">https://link.springer.com/chapter/10.1007/978-3-642-24541-1_27</a></li> <li>W. P. Reinhardt & P. L. Walker (2010) “Weierstrass Elliptic and Modular Functions”, <i>NIST Digital Library of Mathematical Functions</i>, §23.5(v). <a href="https://dlmf.nist.gov/23.5#v">https://dlmf.nist.gov/23.5#v</a></li> <li>P. L. Robinson (2019) “The Dixonian elliptic functions”. <a href="https://arxiv.org/abs/1901.04296">https://arxiv.org/abs/1901.04296</a></li> <li><a href="/facts/Hermann_Schwarz/lP1VRdjH">H. A. Schwarz</a> (1869) “Ueber einige Abbildungsaufgaben”. <i>Crelles Journal</i> 1869(70), 105–120. <a href="https://doi.org/10.1515/crll.1869.70.105">http://doi.org/10.1515/crll.1869.70.105</a></li> <li>B. R. Seth & F. P. White (1934) “Torsion of beams whose cross-section is a regular polygon of n sides”. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, 30(2), 139. <a href="https://doi.org/10.1017/s0305004100016558">http://doi.org/10.1017/s0305004100016558</a> </li> <li>D. Shelupsky (1959) “A generalization of the trigonometric functions”. <i>The American Mathematical Monthly</i> 66(10), 879–884. <a href="https://www.jstor.org/stable/2309789">https://www.jstor.org/stable/2309789</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Desmos_(graphing)/QdznIbMH">Desmos</a> plots: <ul><li>Real-valued Dixon elliptic functions <a href="https://www.desmos.com/calculator/5s4gdcnxh2">https://www.desmos.com/calculator/5s4gdcnxh2</a>.</li> <li>Parametrizing the cubic Fermat curve, <a href="https://www.desmos.com/calculator/elqqf4nwas">https://www.desmos.com/calculator/elqqf4nwas</a></li></ul></li> <li><a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">On-Line Encyclopedia of Integer Sequences</a> pages: <ul><li>“Coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the Dixon elliptic function sm(x,0).” <a href="https://oeis.org/A104133">https://oeis.org/A104133</a></li> <li>“Coefficient of x^(3n)/(3n)! in the Maclaurin expansion of the Dixon elliptic function cm(x,0).” <a href="https://oeis.org/A104134">https://oeis.org/A104134</a></li> <li>“Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z).” <a href="https://oeis.org/A197374">https://oeis.org/A197374</a></li></ul></li> <li>Mathematics <a href="/facts/Stack_Exchange/VQvJKJMh">Stack Exchange</a> discussions: <ul><li>“On x 3 + y 3 = z 3 {\displaystyle x^{3}+y^{3}=z^{3}} , the Dixonian elliptic functions, and the Borwein cubic theta functions”, <a href="https://math.stackexchange.com/q/2090523/">https://math.stackexchange.com/q/2090523/</a></li> <li>“doubly periodic functions as tessellations (other than parallelograms)”, <a href="https://math.stackexchange.com/q/35671/">https://math.stackexchange.com/q/35671/</a></li></ul></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Dixon (1890), Dillner (1873). Dillner uses the symbols W = sm , W 1 = cm . {\displaystyle W=\operatorname {sm} ,\ W_{1}=\operatorname {cm} .} <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Dixon (1890), Van Fossen Conrad & Flajolet (2005), Robinson (2019). <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>The mapping for a general regular polygon is described in Schwarz (1869). <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>van Fossen Conrad & Flajolet (2005) p. 6. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Dillner (1873) calls the period 3 w {\displaystyle 3w} . Dixon (1890) calls it 3 λ {\displaystyle 3\lambda } ; Adams (1925) and Robinson (2019) each call it 3 K {\displaystyle 3K} . Van Fossen Conrad & Flajolet (2005) call it π 3 {\displaystyle \pi _{3}} . Also see OEIS A197374. <a href="https://oeis.org/A197374" target="_blank">https://oeis.org/A197374</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Dixon (1890), Van Fossen Conrad & Flajolet (2005) <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Dixon (1890), Adams (1925) <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Dixon (1890), p. 185–186. Robinson (2019). <a href="https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify=%7B%22pages%22%3A%5B197%5D%7D" target="_blank">https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify=%7B%22pages%22%3A%5B197%5D%7D</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Adams (1925) <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>van Fossen Conrad & Flajolet (2005). Also see OEIS A104133, A104134. <a href="https://oeis.org/A104133" target="_blank">https://oeis.org/A104133</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Reinhardt & Walker (2010) <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Chapling (2018), Robinson (2019). Adams (1925) instead expresses the Dixon elliptic functions in terms of the Weierstrass elliptic function ℘ ( z ; 0 , − 1 ) . {\displaystyle \wp (z;0,-1).} <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>van Fossen Conrad & Flajolet (2005), p.38 <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Lundberg (1879), Grammel (1948), Shelupsky (1959), Burgoyne (1964), Gambini, Nicoletti, & Ritelli (2021). <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Adams (1925), Cox (1935), Magis (1938), Lee (1973), Lee (1976), McIlroy (2011), Chapling (2016). <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>