Burnside Curve
المؤلف:
Brezhnev, Y. V.
المصدر:
"Uniformisation: On the Burnside Curve y^2=x^5-x." 9 Dec 2001. https://arxiv.org/abs/math.CA/0111150.
الجزء والصفحة:
...
6-7-2020
1034
Burnside Curve
The only known classically known algebraic curve of curve genus 
that has an explicit parametrization 
in terms of standard special functions (Burnside 1893, Brezhnev 2001). This equation is given by
 |
(1)
|
The closed portion of the curve has area
where 
is a gamma function.
The closed portion of this curve has a parametrization in terms of the Weierstrass elliptic function given by
where
![f(t)=[P(t)-P(2t)][P(1/2t)-P(t)][P(1/2t)-P(t+2)]×[P(1/2)-P(2t+1)][P(1/2)-P(1)],](https://mathworld.wolfram.com/images/equations/BurnsideCurve/NumberedEquation2.gif) |
(6)
|
the half-periods are given by 
and 
ranges over complex values (Brezhnev 2001).
REFERENCES:
Brezhnev, Y. V. "Uniformisation: On the Burnside Curve 
." 9 Dec 2001. https://arxiv.org/abs/math.CA/0111150.
Burnside, W. S. "Note on the Equation 
." Proc. London Math. Soc. 24, 17-20, 1893.
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