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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

.

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).