parameters {elliptic} | R Documentation |
Calculates the invariants g2 and g3, the e-values e1,e2,e3, and the half periods omega1, omega2, from any one of them.
parameters(Omega=NULL, g=NULL, description=NULL)
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 |
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 0.5 times a (nontrivial) period of minimal
modulus, and omega2 is 0.5 times a period of smallest
modulus having the property omega2/omega1
not real.
The relevant periods are made unique by the further requirement 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 (eg, page 443), and Mathematica, have omega1+omega2+omega3=0 |
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),
where omega3 is defined by
omega1+omega2+omega3=0.
|
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) |
Robin K. S. Hankin
## 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