# parameters

##### Parameters for Weierstrass's P function

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.

- Keywords
- math

##### 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:

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
\(\omega_1+\omega_2+\omega_3=0\)

The nome. Here, \(q=e^{\pi i\omega_2/\omega_1}\).

Complex vector of length 2 holding the invariants

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.

The quantity \(g_2^3-27g_3^2\), often denoted \(\Delta\)

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()`

Boolean, with `TRUE`

corresponding to real
invariants, as per Abramowitz and Stegun

character string indicating which parameter was supplied.
Currently, one of “`o`

” (omega), or “`g`

”
(invariants)

##### Examples

```
# NOT RUN {
## 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
# }
```

*Documentation reproduced from package elliptic, version 1.4-0, License: GPL-2*