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
ω1,ω2
, 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)
|
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 ω1 and ω2 . Notation
follows Chandrasekharan: half period
ω1 is 0.5 times a (nontrivial) period of minimal
modulus, and ω2 is 0.5 times a period of smallest
modulus having the property ω2/ω1
not real.
The relevant periods are made unique by the further requirement that
Re(ω1)>0 , and
Im(ω2)>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 ω3 !
A and S use ω3=ω2−ω1 ,
while Whittaker and Watson (eg, page 443), and Mathematica, have
ω1+ω2+ω3=0
|
q |
The nome. Here,
q=eπiω2/ω1 .
|
g |
Complex vector of length 2 holding the invariants
|
e |
Complex vector of length 3. Here e1 , e2 ,
and e3 are defined by
e1=℘(ω1/2)me2=℘(ω2/2),e3=℘(ω3/2)
where ω3 is defined by
ω1+ω2+ω3=0 .
Note that the e s are also defined as the three roots of
x3−g2x−g3=0 ; but this method cannot be used in
isolation because the roots may be returned in the wrong order.
|
Delta |
The quantity g23−27g32 , often
denoted Δ
|
Eta |
Complex vector of length 3 often denoted
η . Here
η=(η1,η2,η3) are defined
in terms of the Weierstrass zeta function with
ηi=ζ(ωi) 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
[Package
elliptic version 1.4-0
Index]