parameters {elliptic}

Parameters for Weierstrass's P function

Description

Calculates the invariants g_2 and g_3, the e-values e_1,e_2,e_3, and the half periods \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 (\omega_1,\omega_2)

g

Vector of length two: (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 \omega_1 and \omega_2. Notation follows Chandrasekharan: half period \omega_1 is 0.5 times a (nontrivial) period of minimal modulus, and \omega_2 is 0.5 times a period of smallest modulus having the property \omega_2/\omega_1 not real.

The relevant periods are made unique by the further requirement that \mathrm{Re}(\omega_1)>0, and \mathrm{Im}(\omega_2)>0; but note that this often results in sign changes when considering cases on boundaries (such as real g_2 and g_3).

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

q

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

g

Complex vector of length 2 holding the invariants

e

Complex vector of length 3. Here e_1, e_2, and e_3 are defined by

e_1=\wp(\omega1/2)m\qquad e_2=\wp(\omega2/2),\qquad e_3=\wp(\omega3/2)

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

Note that the es are also defined as the three roots of x^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 g_2^3-27g_3^2, often denoted \Delta

Eta

Complex vector of length 3 often denoted \eta. Here \eta=(\eta_1,\eta_2,\eta_3) are defined in terms of the Weierstrass zeta function with \eta_i=\zeta(\omega_i) for i=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