ellip

JAX implementations of various elliptic integrals.

Bulirsch integrals

Forward and reverse mode autodiff compatible

• $el1(x, k_c)$: incomplete elliptic integral of the first kind
• $el2(x, k_c, a, b)$: incomplete elliptic integral of the second kind
• $el3(x, k_c, p)$: incomplete elliptic integral of the third kind
• $cel(k_c, p, a, b)$: Generalized complete elliptic integral

Carlson integrals

Forward mode autodiff only

• $R_\mathrm{F}(x, y, z)$
• $R_\mathrm{C}(x, y)$
• $R_\mathrm{J}(x, y, z, p)$
• $R_\mathrm{D}(x, y, z)$

Legendre forms

Forward mode autodiff only

• $K(k)$: complete elliptic integral of the first kind

• $E(k)$: complete elliptic integral of the second kind

• $\Pi(n, k)$: complete elliptic integral of the third kind

• $F(\phi, k)$: incomplete elliptic integral of the first kind

• $E(\phi, k)$: incomplete elliptic integral of the second kind

• $\Pi(\phi, k, n)$: incomplete elliptic integral of the third kind

Note: The Legendre forms are computed directly from the Carlson integrals using the relations found in [3].

References

[1] Bulirsch, 1969b

[2] Carlson, 1994

[3] DLMF entry on Carlson Symmetric Integrals