Conversions of parameters of Weierstrass \(\wp\)#

Elliptic lattice roots \(e_1, e_2, e_3\) from elliptic lattice invariants \(g_2, g_3\)#

ctxflint.elliptic_roots_from_g2g3(g2, g3)#

Computes the lattice roots \(e_1, e_2, e_3\) from the lattice invariants \(g_2, g_3\), using the equation

\[4 t^3 - g_2 t - g_3 = 4 (t - e_1)(t - e_2)(t - e_3) = 0.\]

See also: MathWorld [292], MathWorld [291], Flint [837].

At least one of \(\boldsymbol{g_2}\) and \(\boldsymbol{g_3}\) is complex

For \(g_2 \cdot g_3 \ne 0\), the roots are determined as the roots of \(4 t^3 - g_2 t - g_3 = 0\) (see CubicEquationRoots).

In general, these roots are complex numbers \(e_1, e_2, e_3\), with \(e_1 + e_2 - e_3 = 0\). They are ordered so that the triangle with vertices \(e_1, e_2, e_3\) is positively oriented and \([e_1, e_3]\) is its longest side (chosen arbitralily if there is more than one). In particular, if \(e_1, e_2, e_3\) are collinear, then we label them so that \(e_2\) is on the line segment \((e_1, e_3)\).

In consequence, \(\displaystyle m_1 = \frac{e_1 - e_3}{e_1 - e_3}, \quad m_2 = \frac{e_2 - e_3}{e_1 - e_3}\) satisfy \(\Im(m_1) \le 0 \le \Im(m_2)\), with strict inequality unless \(e_1, e_2, e_3\) are collinear. Also \(|m_1| \le 1\), \(|m_2| \le 1\), and taking the principal square roots of \(m_1\) and \(m_2\) we obtain values that lie in the fourth and 1st quadrants, respectively.

Both \(\boldsymbol{g_2}\) and \(\boldsymbol{g_3}\) are real

The calculation proceeds depending on the value of the modular delta function \(\Delta = g_2^3 - 27 g_3^2\).

For \(\Delta > 0, g_2 \cdot g_3 \ne 0\), the roots are determined as the roots of \(4 t^3 - g_2 t - g_3 = 0\) (see CubicEquationRoots). All roots are real and are ordered so that \(e_1 > e_2 > e_3\).

For \(\Delta < 0, g_2 \cdot g_3 \ne 0\), the roots are determined as the roots of \(4 t^3 - g_2 t - g_3 = 0\) (see CubicEquationRoots). There is only one real root which is assigned to \(e_2\) and the two other (complex conjugate) roots are assigned to \(e_1\) and \(e_3\), with \(\Im(e_1) > 0\).

For some special cases the lattice roots can be calculated in closed form:

For \(\Delta \ne 0, g_2 > 0, g_3 = 0: \quad e_1 = \tfrac{1}{2} \sqrt{|g_2|}, \quad e_2 = 0, \quad e_3 = -e_1\). (Lemniscate case for \(g_2 = 1\)).

For \(\Delta \ne 0, g_2 < 0, g_3 = 0: \quad e_1 = i \tfrac{1}{2} \sqrt{|g_2|}, \quad e_2 = 0, \quad e_3 = -e_1\). (Pseudo-lemniscate case for \(g_2 = -1\)).

For \(\Delta \ne 0, g_2 = 0, g_3 > 0: \quad e_1 = e_2 \cdot c , \quad e_2 = \sqrt[3]{|g_3|/4}, \quad e_3 = e_2 / c, \quad \text{where } c = e^{2\pi i/3}\). (Equianharmonic case for \(g_3 = 1\)).

For \(\Delta \ne 0, g_2 = 0, g_3 < 0: \quad e_1 = e_2 / c , \quad e_2 = \sqrt[3]{|g_3|/4}, \quad e_3 = e_2 \cdot c, \quad \text{where } c = e^{2\pi i/3}\).

For \(\Delta = 0, g_2 > 0, g_3 > 0: \quad e_1 = 2c, \quad e_2 = e_3 = -c, \quad \text{where } c = \sqrt{g_2/12}\).

For \(\Delta = 0, g_2 > 0, g_3 < 0: \quad e_1 = e_2 = c, \quad e_3 = -2c, \quad \text{where } c = \sqrt{g_2/12}\).

For \(\Delta = 0, g_2 = 0, g_3 = 0: \quad e_1 = e_2 = e_3 = 0\).

An example with purely imaginary input, producing real output:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; tau = '0.0 + 0.7j'
>>> dx2, dx3 = dec.elliptic_invariants(tau); mx2, mx3 = mpm.elliptic_invariants(tau);
>>> gx2, gx3  = gmp.elliptic_invariants(tau)
>>> fx2, fx3 = fpm.elliptic_invariants(tau); ax2, ax3 = apm.elliptic_invariants(tau)
>>> print("lattice invariant g2")
>>> mpm.show([dx2, mx2, gx2, fx2, ax2], aligned=True)
>>> print("lattice invariant g3")
>>> mpm.show([dx3, mx3, gx3, fx3, ax3], aligned=True)

lattice invariant g2
dec: 5.573660602292979232289778942418842671744E+2
mpm: 5.573660602292979232289778942418842671737e+2
gmp: 5.573660602292979232289778942418842671749E+02
fpm: 5.57366060229298E+02
apm: 5.573660602292979232289778942418842671735e+2 (7.909e-38%)

lattice invariant g3
dec: -2.266333763304145743477283974834873543570E+3
mpm: -2.266333763304145743477283974834873543563e+3
gmp: -2.266333763304145743477283974834873543571E+03
fpm: -2.26633376330415E+03
apm: -2.266333763304145743477283974834873543560e+3 (-1.184e-36%)

Elliptic half periods \(\omega_1, \omega_2\) from elliptic lattice invariants \(g_2, g_3\)#

ctxflint.elliptic_halfperiods_from_g2g3(g2, g3)#

Computes the elliptic half periods \(\omega_1, \omega_2\) from the lattice invariants \(g_2, g_3\). Note that in general the pair \((\omega_1, \omega_2)\) is not unique; depending on the input, there are up to 6 different but equivalent solutions, as explained below.

See also: MathWorld [292], MathWorld [291], Flint [837], and https://dlmf.nist.gov/23.2.

This function is intended to enable the use of real and complex elliptic lattice invariants \(g_2 \ne 0\) and \(g_3 \ne 0\) as parameters of WeierstrassP(), WeierstrassPPrime(), WeierstrassPInv(), WeierstrassZeta(), WeierstrassSigma() by providing a direct way to calculate the corresponding values of \(\omega = \omega_1\) and \(\tau = \omega_2 / \omega_1)\), which are the expected parameters of these functions.

At least one of \(\boldsymbol{g_2}\) and \(\boldsymbol{g_3}\) is complex

For \(g_2 \cdot g_3 \ne 0\): we first compute the (ordered) lattice roots \(e_1, e_2, e_3\) from the lattice invariants \(g_2, g_3\) (see EllipticRootsG). Then

\[\omega_1 = \frac{F}{\mathrm{agm}(1,m_1)}, \quad \omega_2 = i \frac{F}{\mathrm{agm}(1, \sqrt{m_2})}, \quad \text{where } m_1 = \frac{e_1 - e_3}{e_1 - e_3}, \quad m_2 = \frac{e_2 - e_3}{e_1 - e_3}, \quad F = \frac{\pi}{3} \sqrt{ \frac{g_2 (2 + m_1 m_2)(m_2 - m_1)}{g_3 (1 - m_1 m_2)}}.\]

This process yields 2 possible pairs \((\omega_1, \omega_2)\), corresponding to the 2 possible choices of the square root; the solution obtained by taking the principal square root is returned.

For two special cases the elliptic half periods \(\omega_1, \omega_2\) can be calculated without calculating the lattice roots first:

For \(\displaystyle g_2 \ne 0, g_3 = 0: \omega_1 = \frac{\Gamma^2(1/4)}{4 \sqrt{\pi} \sqrt[4]{g_2} } , \omega_2 = i \omega_1\). There are 4 possible pairs \((\omega_1, \omega_2)\), corresponding to the 4 rotations of a square lattice; the solution obtained by taking the principal root \((k=0)\) is returned.

For \(\displaystyle g_2 = 0, g_3 \ne 0: \omega_1 = \frac{\Gamma^3(1/3)}{4 \pi \sqrt[6]{g_3} }, \omega_2 = e^{-\pi i/3} \omega_1\). There are 6 possible pairs \((\omega_1, \omega_2)\), corresponding to the 6 rotations of a square lattice of equilateral triangles; the solution obtained by taking the principal root \((k=0)\) is returned.

Both \(\boldsymbol{g_2}\) and \(\boldsymbol{g_3}\) are real

The calculation proceeds depending on the value of the modular delta function \(\Delta = g_2^3 - 27 g_3^2\). \(K(\cdot)\) denotes the complete elliptic integral of the first kind.

For \(\Delta > 0, g_2 \cdot g_3 \ne 0\): we first compute the lattice roots \(e_1, e_2, e_3\) from the lattice invariants \(g_2, g_3\) (see EllipticRootsG). Then

\[\omega_1 = \frac{K(m)}{\sqrt{e_1 - e_3}}, \quad \omega_2 = i \frac{K(1-m)}{\sqrt{e_1 - e_3}}, \quad \text{where } m = \frac{e_2 - e_3}{e_1 - e_3}.\]

For \(\Delta < 0, g_2 \cdot g_3 \ne 0\): we first compute the lattice roots \(e_1, e_2, e_3\) from the lattice invariants \(g_2, g_3\) (see EllipticRootsG). Then

\[\omega_1 = \frac{K(m)}{\sqrt{H_2}}, \quad \omega_2 = i \frac{K(1-m)}{\sqrt{H_2}}, \quad \text{where } m = \tfrac{1}{2} - \frac{3 e_2}{4 H_2}, \quad H_2 = \sqrt{(e_2 - e_1)(e_2 - e_3)}.\]

For some special cases the elliptic half periods \(\omega_1, \omega_2\) can be calculated without calculating the lattice roots first:

For \(\displaystyle \Delta \ne 0, g_2 > 0, g_3 = 0: \quad \omega_1 = \frac{\Gamma^2(1/4)}{4 \sqrt{\pi} \sqrt[4]{|g_2|} } , \quad \omega_2 = i \omega_1\). (Lemniscate case for \(g_2 = 1\)).

For \(\displaystyle \Delta \ne 0, g_2 < 0, g_3 = 0: \quad \omega_1 = \sqrt{2} \frac{\Gamma^2(1/4)}{4 \sqrt{\pi} \sqrt[4]{|g_2|} } , \quad \omega_2 = i \omega_1\). (Pseudo-lemniscate case for \(g_2 = -1\)).

For \(\displaystyle \Delta \ne 0, g_2 = 0, g_3 > 0: \quad \omega_1 = \frac{\Gamma^3(1/3)}{4 \pi \sqrt[6]{|g_3|} }, \quad \omega_2 = i\omega_1 \cdot \sqrt{3}\). (Equianharmonic case for \(g_3 = 1\)).

For \(\displaystyle \Delta \ne 0, g_2 = 0, g_3 < 0: \quad \omega_1 = \sqrt{3} \frac{\Gamma^3(1/3)}{4 \pi \sqrt[6]{|g_3|} }, \quad \omega_2 = i \omega_1 / \sqrt{3}\).

For \(\displaystyle \Delta = 0, g_2 > 0, g_3 > 0: \quad \omega_1 = \frac{\pi}{\sqrt[6]{216|g_3|} }, \quad \omega_2 = i \infty\).

For \(\displaystyle \Delta = 0, g_2 > 0, g_3 < 0: \quad \omega_1 = \infty, \quad \omega_2 = \frac{i \pi}{\sqrt[6]{216|g_3|} }\).

For \(\displaystyle \Delta = 0, g_2 = 0, g_3 = 0: \quad \omega_1 = \infty, \quad \omega_2 = i \infty\).

An example with purely imaginary input, producing real output:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; tau = '0.0 + 0.7j'
>>> dx2, dx3 = dec.elliptic_invariants(tau); mx2, mx3 = mpm.elliptic_invariants(tau);
>>> gx2, gx3  = gmp.elliptic_invariants(tau)
>>> fx2, fx3 = fpm.elliptic_invariants(tau); ax2, ax3 = apm.elliptic_invariants(tau)
>>> print("lattice invariant g2")
>>> mpm.show([dx2, mx2, gx2, fx2, ax2], aligned=True)
>>> print("lattice invariant g3")
>>> mpm.show([dx3, mx3, gx3, fx3, ax3], aligned=True)

lattice invariant g2
dec: 5.573660602292979232289778942418842671744E+2
mpm: 5.573660602292979232289778942418842671737e+2
gmp: 5.573660602292979232289778942418842671749E+02
fpm: 5.57366060229298E+02
apm: 5.573660602292979232289778942418842671735e+2 (7.909e-38%)

lattice invariant g3
dec: -2.266333763304145743477283974834873543570E+3
mpm: -2.266333763304145743477283974834873543563e+3
gmp: -2.266333763304145743477283974834873543571E+03
fpm: -2.26633376330415E+03
apm: -2.266333763304145743477283974834873543560e+3 (-1.184e-36%)

Elliptic half-period \(\omega_1\) and elliptic period ratio \(\tau\) from elliptic lattice invariants \(g_2\) and \(g_3\)#

ctxflint.OmegaAndTau(g2, g3)#

Given the (possibly complex) elliptic lattice invariants \(g_2 \ne 0\) and \(g_3 \ne 0\), with \(\Delta = g_2^3 - 27 g_3^2 \ne 0\), returns the elliptic half-period \(\omega_1\) and elliptic period ratio \(\tau\) as a tuple of complex numbers (\(\omega_1, \tau\)) such that \(g_2 \omega_1^4 = g_{3, \tau}\), \(g_3 \omega_1^6 = g_{3, \tau}\), and \(\displaystyle j(\tau) = 1728 \frac{g_2^3}{g_2^3 - 27 g_3^2}\), where \((g_{2, \tau}, g_{3, \tau})\) are the \((g_2, g_3)\) values returned by calling EllipticInvariants() with \(\tau\) as argument, and \(j(\tau )\) denotes the Klein \(j\)-invariant. The returned tuple (\(\omega_1, \tau\)) is not unique.

See also Wikipedia [1480], MathWorld [1081], NIST [501], Flint [835].

See also: https://dlmf.nist.gov/23.22

This function is intended to enable the use of complex elliptic lattice invariants \(g_2 \ne 0\) and \(g_3 \ne 0\) as parameters of WeierstrassP(), WeierstrassPPrime(), WeierstrassPInv(), WeierstrassZeta(), WeierstrassSigma() by providing a direct way to calculate the corresponding values of \(\omega_1\) and \(\tau\), which are the expected parameters of these functions.

Note that for real elliptic lattice invariants the functions WeierstrassPG(), WeierstrassPPrimeG(), WeierstrassPInvG(), WeierstrassZetaG(), WeierstrassSigmaG() should be used: they are typically faster and more accurate and also support the special cases \(g_2 = 0\), \(g_3 = 0\) and \(\Delta = 0\).

The following algorithm is used: We first calculate (by calling CubicEquationRoots) the 3 roots \(x_k (k = 1,2,3)\) of the cubic equation

\[4 x^3 + (G - 12) x^2 + 12 x -4 = 0, \quad \text{where } G = 27 \frac{g_2^3}{g_2^3 - 27 g_3^2}.\]

For each of these 3 roots \(x_k\) we then calculate (by calling QuadraticEquationRoots) the 2 roots \(\lambda_{k,l} (l = 1, 2)\) of the quadratic equation \(\lambda^2 - \lambda + x_k = 0\), resulting overall in 6 roots \(\lambda_{k,l}\). Calculating

\[\tau_{k,l} = i \frac{\mathrm{agm}(1, \sqrt{1 - \lambda_{k,l}})}{\mathrm{agm}(1, \sqrt{\lambda_{k,l}})}\]

we obtain 6 values \(\tau_{k,l}\) such that \(\displaystyle j(\tau_{k,l}) = 1728 \frac{g_2^3}{g_2^3 - 27 g_3^2}\). Of these \(\tau_{k,l}\) the function returns one with \(g_3 \omega^6 = g_{3, \tau_{k,l}}\), where \(\displaystyle \omega = \sqrt[4]{\frac{g_{2, \tau_{k,l}}}{g_2}}\), and \((g_{2, \tau_{k,l}}, g_{3, \tau_{k,l}})\) are the \((g_2, g_3)\) values returned by calling EllipticInvariants() with \(\tau_{k,l}\) as argument.

An example with purely imaginary input, producing real output:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 40; tau = '0.0 + 0.7j'
>>> \mathrm{d}x = dec.kleinj(tau); mx = mpm.kleinj(tau); gx = gmp.kleinj(tau)
>>> fx = fpm.kleinj(tau); ax = apm.kleinj(tau)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax], aligned=True)
dec: 5.023143714184469499902012736851141207074E+0
mpm: 5.023143714184469499902012736851141207074e+0
gmp: 5.023143714184469499902012736851141207074E+00
fpm: 5.02314371418447E+00
apm: 5.023143714184469499902012736851141207074e+0 (1.554e-38%)

An example with complex input:

>>> from xlcalcnet import dec, mpm, gmp, fpm, apm
>>> mpm.dps = 20; tau = '0.4 + 0.7j'
>>> \mathrm{d}z = dec.kleinj(tau); mz = mpm.kleinj(tau); gz = gmp.kleinj(tau)
>>> fz = fpm.kleinj(tau); az = apm.kleinj(tau)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: -4.1369846595151262778E-2               - 2.6193939695223693294E-1j
mpm: -4.1369846595151262778e-2               - 2.6193939695223693294e-1j
gmp: -4.1369846595151262778E-02              - 2.6193939695223693294E-01j
fpm: -4.13698465951511E-02                   - 2.61939396952237E-01j
apm: -4.1369846595151262777e-2 (-1.358e-16%) - 2.6193939695223693294e-1 (-2.158e-17%)j

Conversions of parameters of Weierstrass \wp

Contents

Conversions of parameters of Weierstrass \(\wp\)#

Elliptic lattice roots \(e_1, e_2, e_3\) from elliptic lattice invariants \(g_2, g_3\)#

Elliptic half periods \(\omega_1, \omega_2\) from elliptic lattice invariants \(g_2, g_3\)#

Elliptic half-period \(\omega_1\) and elliptic period ratio \(\tau\) from elliptic lattice invariants \(g_2\) and \(g_3\)#