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Dixon elliptic function specific values
[edit]
Dixon elliptic functions, are Elliptic functions which parametrize
Fermat curve and are useful for Conformal map projections from Sphere to Triangle-related shapes. It is known that
and
where
denotes set of all Algebraic numbers also
and
where
denotes set of all Origami-constructibles. Where
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Complex specific values
[edit]
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Deriviation methods
[edit]
For one deriviation method, we substitute
and
in sum identities, and make use of reflexion identities
and
to get:[1]
![{\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}}\\[8mu]\operatorname {sm} (x+\omega y)&={\frac {\operatorname {sm} ^{2}x\,\operatorname {cm} y-\omega ^{2}\,\operatorname {cm} x\,\operatorname {sm} ^{2}y}{\operatorname {sm} x\,\operatorname {cm} ^{2}y-\omega \,\operatorname {cm} ^{2}x\,\operatorname {sm} y}}\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/2f32fe71de9b934f63b132726319c86118b84ee9)
- For example:
![{\displaystyle {\begin{aligned}\operatorname {cm} ({\tfrac {1}{6}}\pi _{3}+\omega {\tfrac {1}{3}}\pi _{3})&={\frac {1{\big /}{\sqrt[{3}]{4}}}{-\omega \,{\big /}{\sqrt[{3}]{4}}}}=-\omega ^{2}\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/3a277fb4bd5a3cbcb2b0952aac0ee799961dd009)
Another way to deriviate specific values, is to make use of multiple-argument formulas:[2]
For example, to calculate
, we use cm duplication formula,
![{\displaystyle {\begin{aligned}\operatorname {cm} 2u&={\frac {2\operatorname {cm} ^{3}u-1}{2\operatorname {cm} u-\operatorname {cm} ^{4}u}},\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/3bbb80193a82ee874c446c00f8a470d624569829)
![{\displaystyle {\begin{aligned}\operatorname {cm} ({\tfrac {1}{2}}\pi _{3}+\omega \pi _{3})&=\operatorname {cm} ({\tfrac {1}{2}}\pi _{3})=-1={\frac {2\operatorname {cm} ^{3}({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})-1}{2\operatorname {cm} ({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})-\operatorname {cm} ^{4}({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})}},\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/5b220a68a25ae439820ea2162090b404fe537b75)
Equation
has 4 roots:
![{\displaystyle {\frac {1+{\sqrt {3}}-{\sqrt {2{\sqrt {3}}}}}{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/9fd2099fb89915f7f986939e70f2528587f9be18)
![{\displaystyle {\frac {1+{\sqrt {3}}+{\sqrt {2{\sqrt {3}}}}}{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e373b450f75088c6a1be79c57a4274d70bf76df8)
![{\displaystyle {\frac {1-{\sqrt {3}}-i{\sqrt {2{\sqrt {3}}}}}{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ecf5e3f9acf509d5823e72e973c9a206f7ce6bc7)
![{\displaystyle {\frac {1-{\sqrt {3}}+i{\sqrt {2{\sqrt {3}}}}}{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d6b3de19a263e118a58c0879704fe459607a5657)
- By looking at complex cm domain coloring, we can deduct that
is non-real with positive argument less than
. A complex number has positive argument less than
if and only if it's imaginary part is positive, so:
![{\displaystyle \operatorname {cm} ({\tfrac {1}{4}}\pi _{3}+\omega {\tfrac {1}{2}}\pi _{3})={\frac {1-{\sqrt {3}}+i{\sqrt {2{\sqrt {3}}}}}{2}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/87f67be9e729ca748b49ba42766a2b321a621304)
- ^ Dixon (1890), Adams (1925)
- ^ Dixon (1890), p. 185–186. Robinson (2019).
Generalized Fermat curve trigonometric functions
[edit]
In mathematics, Generalised Fermat curve trigonometric functions are complex functions
which real values parametrize curve
. That's why these functions satisfy the identity
. They are generalizations of regular Trigonometric functions which are the case when
. [1] Generalization of
for other Fermat curves is:
.
Parametrization of Fermat curves
[edit]
are inverses of these integrals:
![{\displaystyle z=\int _{0}^{\sin _{n}z}(1-t^{n})^{\frac {1-n}{n}}dt=\int _{\cos _{n}z}^{1}(1-t^{n})^{\frac {1-n}{n}}dt}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/79099793ddcb9a2b60230deabac74acac296ff98)
They also parametrize
, in a way that the signed area lying between the segment from the origin to
is
for
.
The area in the positive quadrant under the curve
is
.
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Trigonometric functions
[edit]
In case when
, we get Trigonometric functions
and
which satisfy
and parametrize Unit circle.
Reflection identities
[edit]
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Multiple Argument identities
[edit]
Sum and Difference identities
[edit]
Dixon elliptic functions
[edit]
In case when
, we get Dixon elliptic functions
and
which satisfy
with period of
, which parametrize the cubic Fermat curve
.
Let
.
Reflection identities
[edit]
![{\displaystyle {\begin{aligned}\operatorname {cm} {\bar {z}}&={\overline {\operatorname {cm} z}},\\\operatorname {sm} {\bar {z}}&={\overline {\operatorname {sm} z}},\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c2153379ad02076473a4dedd3337c6b560e3cf84)
![{\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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/a65d7d2da22c2219dfbfaf59a9a732c1cd71a552)
![{\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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/97d792d79b33c8e3dc0ec93f419afd5efaa78f03)
![{\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}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/34e96f58bf017945ec09c36fb3264aa0d023eed2)
![{\displaystyle {\begin{aligned}\operatorname {cm} {\bigl (}z+{\tfrac {1}{3}}\omega \pi _{3}{\bigr )}&=\omega ^{2}{\frac {-\operatorname {sm} z}{\operatorname {cm} z}},\\\operatorname {sm} {\bigl (}z+{\tfrac {1}{3}}\omega \pi _{3}{\bigr )}&=\omega {\frac {1}{\operatorname {cm} z}}.\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/2a5373796121815e741b8ab1335ad08aea057b91)
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Multiple Argument identities
[edit]
Sum and Difference identities
[edit]
Quartic Trigonometric functions
[edit]
In case when
, we get
and
which satisfy
with period of
, which parametrize the quartic Fermat curve
. Unlike previous cases, they are not meromorphic, but their squares and ratios are. They are related to Lemniscate elliptic functions by
, where
is hyperbolic lemiscate sine which is related to regular lemniscate functions by:
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- ^ Lundberg (1879), Grammel (1948), Shelupsky (1959), Burgoyne (1964), Gambini, Nicoletti, & Ritelli (2021).