elliptic (version 0.4-13)

parameters: Parameters for Weierstrass's P function

Description

Calculates the invariants $g_2$ and $g_3$, the e-values $e_1,e_2,e_3$, and the half periods $\omega_1,\omega_2$, from any one of them.

Usage

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

Arguments

Omega

Vector of length two, containing the half periods $(\omega_1,\omega_2)$.

g

Vector of length two: $(g_2,g_3)$

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 $\omega_1$ and $\omega_2$. Notation follows Chandrasekharan: half period $\omega_1$ is a (nontrivial) period of minimal modulus, and $\omega_2$ is the next smallest having the property $\omega_2/\omega_1$ not real. It is further required that $Re(\omega_1)>0$, and $Im(\omega_2)>0$; but note that this often results in sign changes when considering cases on boundaries (such as real $g_2$ and $g_3$).

  Note Different definitions exist for $\omega_3$! A and S use $\omega_3=\omega_2-\omega_1$, while Whittaker and Watson have $\omega_1+\omega_2+\omega_3=0$ on, for example, p443.

• q
  The nome. Here, $q=e^{\pi i\omega_2/\omega_1}$.
• g
  Complex vector of length 2 holding the invariants
• e
  Complex vector of length 3. Here $e_1$, $e_2$, and $e_3$ are defined by $$e_1=\wp(\omega1/2)m\qquad e_2=\wp(\omega2/2),\qquad e_3=\wp(\omega3/2)$$
• where $\omega_3$ is defined by $\omega_1+\omega_2+\omega_3=0$.

  Note that the $e$s are also defined as the three roots of $x^3-g_2x-g_3=0$; but this method cannot be used in isolation because the roots may be returned in the wrong order.

• Delta
  The quantity $g_2^3-27g_3^2$, often denoted $\Delta$.
• Eta
  Complex vector of length 3 often denoted $\eta$. Here $\eta=(\eta_1,\eta_2,\eta_3)$ are defined in terms of the Weierstrass zeta function with $\eta_i=\zeta(\omega_i)$ 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).

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