parameters {elliptic}R Documentation

Parameters for Weierstrass's P function

Description

Calculates the invariants g2 and g3, the e-values e1,e2,e3, and the half periods omega1, omega2, from any one of them.

Usage

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

Arguments

Omega Vector of length two, containing the half periods (omega1,omega2).
g Vector of length two: (g2,g3)
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 omega1 and omega2. Notation follows Chandrasekharan: half period omega1 is a (nontrivial) period of minimal modulus, and omega2 is the next smallest having the property omega2/omega1 not real.
It is further required that Re(omega1)>0, and Im(omega2)>0; but note that this often results in sign changes when considering cases on boundaries (such as real g2 and g3).
Note Different definitions exist for omega3! A and S use omega3=omega2-omega1, while Whittaker and Watson have omega1+omega2+omega3=0 on, for example, p443.
q The nome. Here, q=exp(pi*i*omega2/omega1).
g Complex vector of length 2 holding the invariants
e Complex vector of length 3. Here e1, e2, and e3 are defined by

e1=P(omega1/2), e2=P(omega2/2), e3=P(omega3/2),

Delta The quantity g2^3-27*g3^2, often denoted Greek capital Delta.
Eta Complex vector of length 3 often denoted by the greek letter eta. Here eta=(eta_1,eta_2,eta_3) are defined in terms of the Weierstrass zeta function with eta_izeta(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).

Note

note here.

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 0.4-14 Index]