- 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)