Elliptic Functions and Integrals

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Elliptic Functions and Integrals

A brief proposal for providing a complete suite of elliptic functions and integrals as PyTorch operators. Enjoy!

One of a five-part series of special functions issues:

• Bessel and Related Functions (Bessel and Related Functions #76324)
• Elliptic Functions and Integrals (Elliptic Functions and Integrals #80157)
• Gamma and Related Functions (Gamma and Related Functions #78065)
• Orthogonal Polynomials (Orthogonal Polynomials #80152)

Parameterization

More than any other special functions, elliptic functions and integrals are expressed in various ways. In particular, the parameter is expressed using the modulus, $k$, modular angle, \alpha, or the parameter $m$.

The following formulas relate them:

$k = \sin \times \alpha$

$m = k^{2} = \sin \times 2 \times \alpha$

So that the integral of the third kind (for example) may be expressed as either:

$\Pi(n, φ, k)$
$\Pi(n, φ, α)$
$\Pi(n, φ, m)$

This proposal provides k and m variations denoted by either _k or _m suffixes.

API

Elliptic Integrals (k)

• complete_elliptic_integral_k_e(input: Tensor, *, out=None) → Tensor

• complete_elliptic_integral_k_k(input: Tensor, *, out=None) → Tensor

Elliptic Integrals (m)

• complete_elliptic_integral_m_e(input: Tensor, *, out=None) → Tensor

• complete_elliptic_integral_m_k(input: Tensor, *, out=None) → Tensor

Jacobi Elliptic Functions (k)

• jacobi_elliptic_k_amplitude(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_cd(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_cn(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_cs(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_dc(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_dn(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_ds(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_nc(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_nd(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_ns(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_sc(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_sd(input: Tensor, k, *, out=None) → Tensor

• jacobi_elliptic_k_sn(input: Tensor, k, *, out=None) → Tensor

Jacobi Elliptic Functions (m)

• jacobi_elliptic_m_amplitude(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_cd(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_cn(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_cs(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_dc(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_dn(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_ds(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_nc(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_nd(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_ns(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_sc(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_sd(input: Tensor, m, *, out=None) → Tensor

• jacobi_elliptic_m_sn(input: Tensor, m, *, out=None) → Tensor

Inverse Jacobi Elliptic Functions (k)

• inverse_jacobi_elliptic_k_cd(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_cn(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_cs(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_dc(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_dn(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_ds(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_nc(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_nd(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_ns(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_sc(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_sd(input: Tensor, k, *, out=None) → Tensor

• inverse_jacobi_elliptic_k_sn(input: Tensor, k, *, out=None) → Tensor

Inverse Jacobi Elliptic Functions (m)

• inverse_jacobi_elliptic_m_cd(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_cn(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_cs(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_dc(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_dn(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_ds(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_nc(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_nd(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_ns(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_sc(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_sd(input: Tensor, m, *, out=None) → Tensor

• inverse_jacobi_elliptic_m_sn(input: Tensor, m, *, out=None) → Tensor

• carlson_elliptic_integral_rc(x, y)

• carlson_elliptic_integral_rd(x, y, z)

• carlson_elliptic_integral_rf(x, y, z)

• carlson_elliptic_integral_rg(x, y, z)

Jacobi Theta Functions

• jacobi_theta_1(x, q)

• jacobi_theta_2(x, q)

• jacobi_theta_3(x, q)

• jacobi_theta_4(x, q)

Jacobi Theta Derivatives

• jacobi_theta_derivative_1(x, q)

• jacobi_theta_derivative_2(x, q)

• jacobi_theta_derivative_3(x, q)

• jacobi_theta_derivative_4(x, q)