# half.periods

0th

Percentile

##### Calculates half periods in terms of e

Calculates half periods in terms of $e$

Keywords
math
##### Usage
half.periods(ignore=NULL, e=NULL, g=NULL, primitive)
##### Arguments
e

e

g

g

ignore

Formal argument present to ensure that e or g is named (ignored)

primitive

Boolean, with default TRUE meaning to return primitive periods and FALSE to return the direct result of Legendre's iterative scheme

##### 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.

##### References

M. Abramowitz and I. A. Stegun 1965. Handbook of Mathematical Functions. New York, Dover.

• half.periods
##### Examples
# NOT RUN {
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)

# }

Documentation reproduced from package elliptic, version 1.4-0, License: GPL-2

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