Modular forms, in terms of half-period \(\omega_1\) and elliptic period ratio \(\tau\)#

Dedekind eta function \(\eta(\tau)\)#

ctxflint.dedekind_eta(tau)#

Returns Dedekind \(\eta\) in terms of elliptic period ratio \(\tau\). . See also Flint [831], MathWorld [1048], Mathworld, equation 20:

\[\eta(\tau) = \frac{\theta_2(\pi/6, \bar{q}^{1/6})}{\sqrt{3}}, \quad \text{where } \bar{q} = q^2 = e^{2 i \pi \tau}.\]

Returns the Dedekind eta function \(\eta(ix)\) for \(x \ge 0\), with \(\eta(x) = q^{1/24}(q)_{\infty}\), and \((q)_{\infty}\) is is the q-Pochhammer Euler function.

See also: MathWorld [1048], Ehrhardt [309] (3.2.17.10).

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.dedekind_eta(tau); mx = mpm.dedekind_eta(tau); gx = gmp.dedekind_eta(tau)
>>> fx = fpm.dedekind_eta(tau); ax = apm.dedekind_eta(tau)
>>> mpm.show([\mathrm{d}x, mx, gx, fx, ax], aligned=True)
dec: 8.221864477624933414117646494972458581160E-1
mpm: 8.221864477624933414117646494972458581160e-1
gmp: 8.221864477624933414117646494972458581160E-01
fpm: 8.22186447762493E-01
apm: 8.221864477624933356092138235685057376542e-1 (2.094e-39%) + 0.0e+0 (0.0%)j

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.dedekind_eta(tau); mz = mpm.dedekind_eta(tau); gz = gmp.dedekind_eta(tau)
>>> fz = fpm.dedekind_eta(tau); az = apm.dedekind_eta(tau)
>>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
dec: 8.3680777459719110894E-1              + 8.2020616070546877819E-2j
mpm: 8.3680777459719110894e-1              + 8.2020616070546877819e-2j
gmp: 8.3680777459719110894E-01             + 8.2020616070546877819E-02j
fpm: 8.36807774597191E-01                  + 8.20206160705469E-02j
apm: 8.3680777459719109849e-1 (3.543e-19%) + 8.2020616070546884265e-2 (2.84e-18%)j

Elliptic modular lambda function \(\lambda(\tau)\) (also DAMath)#

ctxflint.math53.elliptic_modular_lambda(tau)#

Computes the lambda function \(\lambda(\tau) = \theta_2^4(0,\tau) / \theta_3^4(0,\tau)\) in terms of elliptic period ratio \(\tau\). It is invariant under modular transformations \((a, b; c, d)\) where \(a, d\) are odd and \(b, c\) are even.

Elliptic modular delta function \(\Delta(\omega, \tau)\)#

ctxflint.math53.elliptic_modular_delta(omega, tau)#

Computes the modular discriminant \(\Delta(\tau) = \eta(\tau)^{24}\) in terms of half-period \(\omega_1\) and elliptic period ratio \(\tau\). It transforms as

\[\Delta\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{12} \Delta(\tau).\]

The modular discriminant is sometimes defined with an extra factor \((2\pi)^{12}\), which we omit in this implementation.

We have \(\Delta(t \omega, \tau) = t^{-12} \Delta(\omega, \tau)\).

\[\Delta =g_{2}^{3}-27g_{3}^{2} = 4096\pi ^{12}\eta (\tau )^{24}\]

for Weierstrass invariants \(g_2, g_3\), and Dedekind eta function \(\eta(\tau)\).

See also Wikipedia [1463], MathWorld [1064], Flint [835].

Klein j-invariant \(j(\tau )\) (also DAMath)#

ctxflint.klein_j(tau)#

Returns the Klein \(j\)-invariant in terms of elliptic period ratio \(\tau\). See also Wikipedia [1480], MathWorld [1081], NIST [501], Flint [835], Mpmath [723].

Computes Klein’s \(j\)-invariant \(j(\tau)\) given \(\tau\) in the upper half-plane. The function is normalized so that \(j(i) = 1728\). We first move \(\tau\) to the fundamental domain, which does not change the value of the function. Then we use the formula

\[j(\tau ) = 1728{\frac {g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}} = 32 (\theta_2^8+\theta_3^8+\theta_4^8)^3 / (\theta_2 \theta_3 \theta_4)^8\]

where \(\theta_i = \theta_i(0,\tau)\).

Returns the Klein j-invariant \(J(\tau), \tau = iy, y \ge 0\), \(\displaystyle J(\tau) = \frac{\left(\theta_2^8(q)+\theta_3^8(q)+\theta_4^8(q)\right)^3}{54 (\theta'_1)^8(q)}, q = e^{i \pi \tau} = e^{-\pi y}\).

See also: Wikipedia [1480], MathWorld [1081], Ehrhardt [309] (3.2.17.12).

Returns the Klein j-invariant in terms of elliptic period ratio \(\tau\). See also Wikipedia [1480], MathWorld [1081], NIST [501], Flint [835].

\[j(g_1, g_2) = 1728{\frac {g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}}\]