half.periods {elliptic} | R Documentation |
Calculates half periods in terms of e
Description
Calculates half periods in terms of e
Usage
half.periods(ignore=NULL, e=NULL, g=NULL, primitive)
Arguments
e |
e |
g |
g |
ignore |
Formal argument present to ensure that |
primitive |
Boolean, with default |
Details
Parameter e=c(e1,e2,e3)
are the values of the Weierstrass
\wp
function at the half periods:
e_1=\wp(\omega_1)\qquad e_2=\wp(\omega_2)\qquad e_3=
\wp(\omega_3)
where
\omega_1+\omega_2+\omega_3=0.
Also, e
is given by the roots of the cubic
equation x^3-g_2x-g_3=0
, but the problem is
finding which root corresponds to which of the three elements of
e
.
Value
Returns a pair of primitive half periods
Note
Function parameters()
uses function half.periods()
internally, so do not use parameters()
to determine e
.
Author(s)
Robin K. S. Hankin
References
M. Abramowitz and I. A. Stegun 1965. Handbook of Mathematical Functions. New York, Dover.
Examples
half.periods(g=c(8,4)) ## Example 6, p665, LHS
u <- half.periods(g=c(-10,2))
massage(c(u[1]-u[2] , u[1]+u[2])) ## Example 6, p665, RHS
half.periods(g=c(10,2)) ## Example 7, p665, LHS
u <- half.periods(g=c(7,6))
massage(c(u[1],2*u[2]+u[1])) ## Example 7, p665, RHS
half.periods(g=c(1,1i, 1.1+1.4i))
half.periods(e=c(1,1i, 2, 1.1+1.4i))
g.fun(half.periods(g=c(8,4))) ## should be c(8,4)