elliptix 1.1.1

A library for elliptic functions

Elliptix

A library of algorithms for evaluating elliptic functions of a complex argument.

Installation

To install Elliptix, run the following command in your terminal:

pip install elliptix

Documentation

As of right now, the documentation is handled through function Docstrings in the source file.

Implemented Functions

— The Jacobi Theta functions: $\theta_n(z, q) ; n=1, 2, 3, 4$

jacobi_theta(n, z, q)

— The natural logarithm of the Jacobi Theta functions: $\log(\theta_n(z, q)) ; n=1, 2, 3, 4$

log_jacobi_theta(n, z, q)

— The Weierstrass elliptic functions: $\wp(z; g_2, g_3), \wp'(z; g_2, g_3), \wp^{-1}(w; g_2, g_3), \sigma(z; g_2, g_3), \zeta(z; g_2, g_3)$

weierstrass_p(z, g2, g3)
weierstrass_p_prime(z, g2, g3)
inverse_weierstrass_p(z, g2, g3)
weierstrass_sigma(z, g2, g3)
weierstrass_zeta(z, g2, g3)

— The conversion functions between the modulus $k,$ the parameter $m,$ the half-period ratio $\tau,$ the nome $q,$ the lattice roots $e_1, e_2, e_3,$ the invariants $g_2, g_3,$ the half-periods $\omega_1, \omega_2, \omega_3,$ and the values of the Weierstrass $\zeta$ function at the half-periods $\eta_1, \eta_2, \eta_3$

modulus_k(...)
parameter_m(...)
half_period_ratio_tau(...)
nome_q(...)
weierstrass_e(...)
weierstrass_g(...)
weierstrass_w(...)
weierstrass_eta(...)

— Modular functions: $J(\tau)$, $J^{-1}(\tau)$, $\lambda(\tau)$, $\eta(\tau)$, $\eta'(\tau)$, $G_n(q)$, $E_n(q)$, $\phi(q)$

klein_j(tau)
inverse_klein_j(tau)
modular_lambda(tau)
dedekind_eta(tau)
dedekind_eta_prime(tau)
eisenstein_g(n, q)
eisenstein_e(n, q)
euler_phi(q)

— The Jacobi elliptic functions: $\text{sn}(u, m), \text{cn}(u, m), \text{dn}(u, m), \text{am}(u, m), ...$ and their inverses: $\text{sn}^{-1}(w, m), \text{cn}^{-1}(w, m), ...$

jacobi_ellipfun("sn", u, m)
jacobi_ellipfun("cn", u, m)
jacobi_ellipfun("dn", u, m)
jacobi_ellipfun("am", u, m)
...
inverse_jacobi_ellipfun("sn", u, m)
inverse_jacobi_ellipfun("cn", u, m)
...

— The Neville Theta functions: $\theta_c(u, m), \theta_d(u, m), \theta_n(u, m), \theta_s(u, m)$

neville_theta_c(u, m)
neville_theta_d(u, m)
neville_theta_n(u, m)
neville_theta_s(u, m)

— The Lemniscate elliptic functions: $\text{sinlem}(z), \text{coslem}(z), \text{sinhlem}(z), \text{coshlem}(z)$ and their inverses: $\text{arcsinlem}(z), \text{arccoslem}(z), \text{arcsinhlem}(z), \text{arccoshlem}(z)$

sinlem(z)
coslem(z)
sinhlem(z)
arcsinlem(z)
arcsinhlem(z)
...

Complex Plots

Plotting the elliptic functions on the complex plane using domain coloring reveals beautiful repeating patterns. All functions were plotted using elliptix.plot.ComplexPlot at 1500x1500 pixel resolution.

The Weierstrass ℘ function.

The Weierstrass σ function.

The Jacobi sc function.

The Eisenstein G₆ series as a function of q.

The Eisenstein E₁₂ series as a function of τ.

The Absolute Klein J invariant as a function of τ.

The Dedekind η function as a function of τ.

The Modular λ function as a function of the nome, q.

Citation

@misc{elliptix,
  author       = {Rudolf Rosendorf},
  title        = {Elliptix: Algorithms for elliptic functions},
  month        = {June},
  year         = {2026},
  publisher    = {GitHub},
  journal      = {GitHub repository},
  howpublished = {\url{https://github.com/Ruda975/elliptix}}
}

References

• [DLMF] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.6 of 2026-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

• Weisstein, E. W. (n.d.). MathWorld. Wolfram Research. https://mathworld.wolfram.com/.

• The Mathematical Functions Site. Wolfram Research, Inc., https://functions.wolfram.com/.