Such functions were first introduced to study hyperelliptic integrals, i . e . for the case where S is a hyperelliptic curve.
32.
An application of Hersch's formula to the hyperelliptic quotient of the Riemann surface proves the filling area conjecture in this case.
33.
However, application of this formula in practice is difficult because the relevant hyperelliptic integrals and higher genus theta functions are very complex.
34.
Nonetheless his research continued; his seminal work on hyperelliptic sigma functions dates from around this period, being published in 1886 and 1888.
35.
The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in " x ".
36.
For information about other possible operations on elliptic curves see http : / / hyperelliptic . org / EFD / g1p / index . html.
37.
This statement about genus remains true for " g " = 0 or 1, but those curves are not called " hyperelliptic ".
38.
The gonality is 2 for curves of genus 1 ( elliptic curves ) and for hyperelliptic curves ( this includes all curves of genus 2 ).
39.
Later in his career, Coble also studied the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface.
40.
Basic aspects of Teichmuller theory as applied to something Newton might have been interested; e . g ., something on Igusa invariants of hyperelliptic curves.
