Parameters for Weierstrass's P function — parameters • elliptic

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.

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 0.5 times a (nontrivial) period of minimal modulus, and $\omega_2$ is 0.5 times a period of smallest modulus having the property $\omega_2/\omega_1$ not real.

The relevant periods are made unique by the further requirement that $\mathrm{Re}(\omega_1)>0$, and $\mathrm{Im}(\omega_2)>0$; 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 (eg, page 443), and Mathematica, have $\omega_1+\omega_2+\omega_3=0$

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 $$e_1=\wp(\omega_1/2)m\qquad e_2=\wp(\omega_2/2),\qquad e_3=\wp(\omega_3/2)$$ 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)

Author

Robin K. S. Hankin

Examples

 ## Example 6, p665, LHS
 parameters(g=c(10,2))
#> $Omega
#> [1] 1.003847+0.00000i 0.000000+1.09797i
#> attr(,"class")
#> [1] "primitive"
#> 
#> $q
#> [1] 0.03218824+0i
#> 
#> $e
#>         e1         e2         e3 
#>  1.6729816 -0.2033642 -1.4696174 
#> 
#> $g
#> [1] 10  2
#> 
#> $Delta
#> [1] 892
#> 
#> $Eta
#> [1]  0.7988783+0.0000000i  0.0000000-0.6909934i -0.7988783+0.6909934i
#> 
#> $is.AnS
#> [1] FALSE
#> 
#> $given
#> [1] "g"
#> 
#> attr(,"class")
#> [1] "parameters"
 ## Not clear to me how AMS-55 justify 7 sig figs

 ## Example 7, p665, RHS
 a <- parameters(g=c(7,6)) ;  attach(a)
 c(omega2=Omega[1],omega2dash=Omega[1]+Omega[2]*2)
#>                     omega2                 omega2dash 
#> 9.957998e-01+2.220446e-16i 1.110223e-16+2.332418e+00i


  ## verify 18.3.37:
  Eta[2]*Omega[1]-Eta[1]*Omega[2]   #should be close to pi*1i/2
#> [1] 0-1.570796i


## from Omega to g and and back;
## following should be equivalent to c(1,1i):
 parameters(g=parameters(Omega=c(1,1i))$g)$Omega 
#> [1] 1.83518e-30-1.000000e+00i 1.00000e+00+1.840095e-30i
#> attr(,"class")
#> [1] "primitive"