Description Usage Arguments Value Author(s) Examples
Calculates the invariants g2 and g3, the evalues e1,e2,e3, and the half periods omega1, omega2, from any one of them.
1  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=omega2omega1, 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 ω1+omega2+omega3=0. Note that the es are also defined as the three roots of x^3g2*xg3=0; but this method cannot be used in isolation because the roots may be returned in the wrong order. 
Delta 
The quantity g2^327*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_iζ(omega_i) for i=1,2,3. Note that the name of this element is capitalized to avoid confusion
with function 
is.AnS 
Boolean, with 
given 
character string indicating which parameter was supplied.
Currently, one of “ 
Robin K. S. Hankin
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  ## 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

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