parameters {elliptic}R Documentation

Parameters for Weierstrass's P function

Description

Calculates the invariants g2g_2 and g3g_3, the e-values e1,e2,e3e_1,e_2,e_3, and the half periods ω1,ω2\omega_1,\omega_2, from any one of them.

Usage

parameters(Omega=NULL, g=NULL, description=NULL)

Arguments

Omega

Vector of length two, containing the half periods (ω1,ω2)(\omega_1,\omega_2)

g

Vector of length two: (g2,g3)(g_2,g_3)

description

string containing “equianharmonic”, “lemniscatic”, or “pseudolemniscatic”, to specify one of A and S's special cases

Value

Returns a list with the following items:

Omega

A complex vector of length 2 giving the fundamental half periods ω1\omega_1 and ω2\omega_2. Notation follows Chandrasekharan: half period ω1\omega_1 is 0.5 times a (nontrivial) period of minimal modulus, and ω2\omega_2 is 0.5 times a period of smallest modulus having the property ω2/ω1\omega_2/\omega_1 not real.

The relevant periods are made unique by the further requirement that Re(ω1)>0\mathrm{Re}(\omega_1)>0, and Im(ω2)>0\mathrm{Im}(\omega_2)>0; but note that this often results in sign changes when considering cases on boundaries (such as real g2g_2 and g3g_3).

Note Different definitions exist for ω3\omega_3! A and S use ω3=ω2ω1\omega_3=\omega_2-\omega_1, while Whittaker and Watson (eg, page 443), and Mathematica, have ω1+ω2+ω3=0\omega_1+\omega_2+\omega_3=0

q

The nome. Here, q=eπiω2/ω1q=e^{\pi i\omega_2/\omega_1}.

g

Complex vector of length 2 holding the invariants

e

Complex vector of length 3. Here e1e_1, e2e_2, and e3e_3 are defined by

e1=(ω1/2)me2=(ω2/2),e3=(ω3/2)e_1=\wp(\omega1/2)m\qquad e_2=\wp(\omega2/2),\qquad e_3=\wp(\omega3/2)

where ω3\omega_3 is defined by ω1+ω2+ω3=0\omega_1+\omega_2+\omega_3=0.

Note that the ees are also defined as the three roots of x3g2xg3=0x^3-g_2x-g_3=0; but this method cannot be used in isolation because the roots may be returned in the wrong order.

Delta

The quantity g2327g32g_2^3-27g_3^2, often denoted Δ\Delta

Eta

Complex vector of length 3 often denoted η\eta. Here η=(η1,η2,η3)\eta=(\eta_1,\eta_2,\eta_3) are defined in terms of the Weierstrass zeta function with ηi=ζ(ωi)\eta_i=\zeta(\omega_i) for i=1,2,3i=1,2,3.

Note that the name of this element is capitalized to avoid confusion with function eta()

is.AnS

Boolean, with TRUE corresponding to real invariants, as per Abramowitz and Stegun

given

character string indicating which parameter was supplied. Currently, one of “o” (omega), or “g” (invariants)

Author(s)

Robin K. S. Hankin

Examples

 ## Example 6, p665, LHS
 parameters(g=c(10,2+0i))


 ## Example 7, p665, RHS
 a <- parameters(g=c(7,6)) ;  attach(a)
 c(omega2=Omega[1],omega2dash=Omega[1]+Omega[2]*2)


  ## verify 18.3.37:
  Eta[2]*Omega[1]-Eta[1]*Omega[2]   #should be close to pi*1i/2


## from Omega to g and and back;
## following should be equivalentto c(1,1i):
 parameters(g=parameters(Omega=c(1,1i))$g)$Omega 



[Package elliptic version 1.4-0 Index]