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 cm 3 z + sm 3 z = 1 {\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1} , as real functions they parametrize the cubic Fermat curve x 3 + y 3 = 1 {\displaystyle x^{3}+y^{3}=1} , just as the trigonometric functions sine and cosine parametrize the unit circle x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} .
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.
Definition
The functions sm and cm can be defined as the solutions to the initial value problem:2
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}Or as the inverse of the Schwarz–Christoffel mapping from the complex unit disk to an equilateral triangle, the Abelian integral:3
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}}}}which can also be expressed using the hypergeometric function:4
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 )}}Parametrization of the cubic Fermat curve
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 beta function and Γ {\displaystyle \Gamma } the gamma function:5
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}}}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 Fermat curve 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.6 To see why, apply Green's theorem:
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.}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}} :
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}}}Symmetries
The function sm z {\displaystyle \operatorname {sm} z} has zeros 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 root of unity, 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 Eisenstein integer). 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 poles 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 )} .
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} } .
Fundamental reflections, rotations, and translations
Both cm and sm commute with complex conjugation,
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}}}Analogous to the parity of trigonometric functions (cosine an even function and sine an odd function), 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:
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}}}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},}
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}}}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,
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}}}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
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}}}Specific values
| z {\displaystyle z} | cm z {\displaystyle \operatorname {cm} z} | sm z {\displaystyle \operatorname {sm} z} |
|---|---|---|
| − 1 3 π 3 {\displaystyle {-{\tfrac {1}{3}}}\pi _{3}} | ∞ {\displaystyle \infty } | ∞ {\displaystyle \infty } |
| − 1 6 π 3 {\displaystyle {-{\tfrac {1}{6}}}\pi _{3}} | 2 3 {\displaystyle {\sqrt[{3}]{2}}} | − 1 {\displaystyle -1} |
| 0 {\displaystyle 0} | 1 {\displaystyle 1} | 0 {\displaystyle 0} |
| 1 6 π 3 {\displaystyle {\tfrac {1}{6}}\pi _{3}} | 1 / 2 3 {\displaystyle 1{\big /}{\sqrt[{3}]{2}}} | 1 / 2 3 {\displaystyle 1{\big /}{\sqrt[{3}]{2}}} |
| 1 3 π 3 {\displaystyle {\tfrac {1}{3}}\pi _{3}} | 0 {\displaystyle 0} | 1 {\displaystyle 1} |
| 1 2 π 3 {\displaystyle {\tfrac {1}{2}}\pi _{3}} | − 1 {\displaystyle -1} | 2 3 {\displaystyle {\sqrt[{3}]{2}}} |
| 2 3 π 3 {\displaystyle {\tfrac {2}{3}}\pi _{3}} | ∞ {\displaystyle \infty } | ∞ {\displaystyle \infty } |
More specific values
| z {\displaystyle z} | cm z {\displaystyle \operatorname {cm} z} | sm z {\displaystyle \operatorname {sm} z} |
|---|---|---|
| − 1 4 π 3 {\displaystyle {-{\tfrac {1}{4}}}\pi _{3}} | 1 + 3 + 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}} | − 1 − 3 + 2 3 4 3 {\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} |
| − 2 9 π 3 {\displaystyle -{\tfrac {2}{9}}\pi _{3}} | 3 6 2 sin ( 1 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}} | − 2 cos ( 1 18 π ) 3 6 {\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt[{6}]{3}}}} |
| − 1 9 π 3 {\displaystyle -{\tfrac {1}{9}}\pi _{3}} | 2 sin ( 2 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt[{6}]{3}}}} | − 3 6 2 cos ( 1 18 π ) {\displaystyle -{\frac {\sqrt[{6}]{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}} |
| − 1 12 π 3 {\displaystyle -{\tfrac {1}{12}}\pi _{3}} | − 1 + 3 + 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} | − 1 + 3 − 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} |
| 1 12 π 3 {\displaystyle {\tfrac {1}{12}}\pi _{3}} | − 1 + 3 + 2 3 4 3 {\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} | 1 + 3 − 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}} |
| 1 9 π 3 {\displaystyle {\tfrac {1}{9}}\pi _{3}} | 3 6 2 sin ( 2 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}} | 2 sin ( 1 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt[{6}]{3}}}} |
| 2 9 π 3 {\displaystyle {\tfrac {2}{9}}\pi _{3}} | 2 sin ( 1 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {1}{9}}\pi \right)}{\sqrt[{6}]{3}}}} | 3 6 2 sin ( 2 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {2}{9}}\pi \right)}}} |
| 1 4 π 3 {\displaystyle {\tfrac {1}{4}}\pi _{3}} | 1 + 3 − 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}} | − 1 + 3 + 2 3 4 3 {\displaystyle {\frac {-1+{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} |
| 5 12 π 3 {\displaystyle {\tfrac {5}{12}}\pi _{3}} | − 1 + 3 − 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} | − 1 + 3 + 2 3 2 2 3 {\displaystyle {\frac {-1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2{\sqrt[{3}]{2}}}}} |
| 4 9 π 3 {\displaystyle {\tfrac {4}{9}}\pi _{3}} | − 3 6 2 cos ( 1 18 π ) {\displaystyle -{\frac {\sqrt[{6}]{3}}{2\cos \left({\frac {1}{18}}\pi \right)}}} | 2 sin ( 2 9 π ) 3 6 {\displaystyle {\frac {2\sin \left({\frac {2}{9}}\pi \right)}{\sqrt[{6}]{3}}}} |
| 5 9 π 3 {\displaystyle {\tfrac {5}{9}}\pi _{3}} | − 2 cos ( 1 18 π ) 3 6 {\displaystyle -{\frac {2\cos \left({\frac {1}{18}}\pi \right)}{\sqrt[{6}]{3}}}} | 3 6 2 sin ( 1 9 π ) {\displaystyle {\frac {\sqrt[{6}]{3}}{2\sin \left({\frac {1}{9}}\pi \right)}}} |
| 7 12 π 3 {\displaystyle {\tfrac {7}{12}}\pi _{3}} | − 1 − 3 + 2 3 4 3 {\displaystyle {\frac {-1-{\sqrt {3+2{\sqrt {3}}}}}{\sqrt[{3}]{4}}}} | 1 + 3 + 2 3 2 {\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}} |
Sum and difference identities
The Dixon elliptic functions satisfy the argument sum and difference identities:7
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}}}These formulas can be used to compute the complex-valued functions in real components:
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}}}Multiple-argument identities
Argument duplication and triplication identities can be derived from the sum identity:8
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}}}Specific value identities
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}}}
where cl {\displaystyle \operatorname {cl} } is lemniscate cosine and ϖ {\displaystyle \varpi } is Lemniscate constant.
Power series
The cm and sm functions can be approximated for | z | < 1 3 π 3 {\displaystyle |z|<{\tfrac {1}{3}}\pi _{3}} by the Taylor series
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}}}whose coefficients satisfy the recurrence c 0 = s 0 = 1 , {\displaystyle c_{0}=s_{0}=1,} 9
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}}}These recurrences result in:10
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}}}Relation to other elliptic functions
Weierstrass elliptic function
The equianharmonic Weierstrass elliptic function ℘ ( z ) = ℘ ( z ; 0 , 1 27 ) , {\displaystyle \wp (z)=\wp {\bigl (}z;0,{\tfrac {1}{27}}{\bigr )},} with lattice 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:11
℘ ( 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)}The function ℘ ( z ) {\displaystyle \wp (z)} solves the differential equation:
℘ ′ ( z ) 2 = 4 ℘ ( z ) 3 − 1 27 {\displaystyle \wp '(z)^{2}=4\wp (z)^{3}-{\tfrac {1}{27}}}We can also write it as the inverse of the integral:
z = ∫ ∞ ℘ ( z ) d w 4 w 3 − 1 27 {\displaystyle z=\int _{\infty }^{\wp (z)}{\frac {dw}{\sqrt {4w^{3}-{\tfrac {1}{27}}}}}}In terms of ℘ ( z ) {\displaystyle \wp (z)} , the Dixon elliptic functions can be written:12
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}}}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:
℘ ′ ( 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)}}}Jacobi elliptic functions
The Dixon elliptic functions can also be expressed using Jacobi elliptic functions, which was first observed by Cayley.13 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
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}}.}Finally, the Dixon elliptic functions are as so:
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).}Generalized trigonometry
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.14
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:
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}}}}The area in the positive quadrant under the curve x n + y n = 1 {\displaystyle x^{n}+y^{n}=1} is
∫ 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}}.}The quartic n = 4 {\displaystyle n=4} 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.15
See also
- Eisenstein integer
- Elliptic function
- Lee conformal world in a tetrahedron
- Schwarz–Christoffel mapping
Notes
- 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 ∫ d x / ( 1 − x 3 ) 2 / 3 {\textstyle \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 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.
- A. Dixon (1894) The elementary properties of the elliptic functions. MacMillian. https://archive.org/details/elempropellipt00dixorich/
- Van Fossen Conrad, Eric; Flajolet, Philippe (2005). "The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion". Séminaire Lotharingien de Combinatoire. 54: Art. B54g, 44. arXiv:math/0507268. Bibcode:2005math......7268V. MR 2223029.
- 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
- J. C. Langer & D. A. Singer (2014) “The Trefoil”. Milan Journal of Mathematics 82(1), 161–182. https://case.edu/artsci/math/langer/jlpreprints/Trefoil.pdf
- M. Laurent (1949) “Tables de la fonction elliptique de Dixon pour l’intervalle 0-0, 1030”. Bulletin de l’Académie Royale des Sciences de Belgique Classe des Sciences, 35, 439–450.
- 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) 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 The Canadian Cartographer 13.
- E. Lundberg (1879) “Om hypergoniometriska funktioner af komplexa variabla”. Manuscript, 1879. Translation by Jaak Peetre “On hypergoniometric functions of complex variables”. https://web.archive.org/web/20161024183030/http://www.maths.lth.se/matematiklu/personal/jaak/hypergf.ps
- J. Magis (1938) “Calcul du canevas de la représentation conforme de la sphère entière dans un triangle équilatéral”. Bulletin Géodésique 59(1), 247–256. http://doi.org/10.1007/BF03029866
- M. D. McIlroy (2011) “Wallpaper maps”. Dependable and Historic Computing. Springer. 358–375. https://link.springer.com/chapter/10.1007/978-3-642-24541-1_27
- W. P. Reinhardt & P. L. Walker (2010) “Weierstrass Elliptic and Modular Functions”, NIST Digital Library of Mathematical Functions, §23.5(v). https://dlmf.nist.gov/23.5#v
- P. L. Robinson (2019) “The Dixonian elliptic functions”. https://arxiv.org/abs/1901.04296
- H. A. Schwarz (1869) “Ueber einige Abbildungsaufgaben”. Crelles Journal 1869(70), 105–120. http://doi.org/10.1515/crll.1869.70.105
- B. R. Seth & F. P. White (1934) “Torsion of beams whose cross-section is a regular polygon of n sides”. Mathematical Proceedings of the Cambridge Philosophical Society, 30(2), 139. http://doi.org/10.1017/s0305004100016558
- D. Shelupsky (1959) “A generalization of the trigonometric functions”. The American Mathematical Monthly 66(10), 879–884. https://www.jstor.org/stable/2309789
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 x 3 + y 3 = z 3 {\displaystyle 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/
References
Dixon (1890), Dillner (1873). Dillner uses the symbols W = sm , W 1 = cm . {\displaystyle W=\operatorname {sm} ,\ W_{1}=\operatorname {cm} .} ↩
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 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. https://oeis.org/A197374 ↩
Dixon (1890), Van Fossen Conrad & Flajolet (2005) ↩
Dixon (1890), Adams (1925) ↩
Dixon (1890), p. 185–186. Robinson (2019). https://gdz.sub.uni-goettingen.de/id/PPN600494829_0024?tify=%7B%22pages%22%3A%5B197%5D%7D ↩
Adams (1925) ↩
van Fossen Conrad & Flajolet (2005). Also see OEIS A104133, A104134. https://oeis.org/A104133 ↩
Reinhardt & Walker (2010) ↩
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).} ↩
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). ↩