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Dixon elliptic function specific values

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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  

Simple real values

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Complex specific values

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Deriviation methods

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For one deriviation method, we substitute   and   in sum identities, and make use of reflexion identities   and   to get:[1]

 
For example:
 

Another way to deriviate specific values, is to make use of multiple-argument formulas:[2]

For example, to calculate  , we use cm duplication formula,

 
 

Equation   has 4 roots:

 
 
 
 
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:
 
  1. ^ Dixon (1890), Adams (1925)
  2. ^ Dixon (1890), p. 185–186. Robinson (2019).

Generalized Fermat curve trigonometric functions

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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

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  are inverses of these integrals:

 

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

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In case when  , we get Trigonometric functions   and   which satisfy   and parametrize Unit circle.

Reflection identities

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Specific Values

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Multiple Argument identities

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Sum and Difference identities

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Derivatives

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Dixon elliptic functions

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In case when  , we get Dixon elliptic functions   and   which satisfy   with period of  , which parametrize the cubic Fermat curve  .

Let  .

Reflection identities

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Specific Values

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Multiple Argument identities

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Sum and Difference identities

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Derivatives

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Quartic Trigonometric functions

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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: 

Specific Values

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  1. ^ Lundberg (1879), Grammel (1948), Shelupsky (1959), Burgoyne (1964), Gambini, Nicoletti, & Ritelli (2021).