# parameters: Parameters for Weierstrass's P function In elliptic: Weierstrass and Jacobi Elliptic Functions

## 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

 `1` ```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 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 ω1+omega2+omega3=0. Note that the es are also defined as the three roots of x^3-g2*x-g3=0; but this method cannot be used in isolation because the roots may be returned in the wrong order. `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_iζ(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

 ``` 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,omega2dash=Omega+Omega*2) ## verify 18.3.37: Eta*Omega-Eta*Omega #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 ```

elliptic documentation built on May 2, 2019, 9:37 a.m.