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Crate ellip

Crate ellip 

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§Ellip - Elliptic integrals for Rust

Ellip is a pure-Rust implementation of elliptic integrals. This means there is no dependence on C++ libraries. Ellip also provides less common functions like Bulirsch’s cel and el. Some applications of elliptic integrals include computing the lengths of plane curves, magnetic field from magnets of various shapes (e.g., cylindrical), astrophysics, and string theory.

Use Ellip-Rayon to parallelize and improve performance for large inputs. Ellip is also available for Python via EllipPy.

§Example

Calculating the perimeter of an ellipse.

use ellip::*;

fn ellipse_perimeter(a: f64, b: f64) -> Result<f64, StrErr> {
    Ok(8.0 * elliprg(0.0, a * a, b * b)?)
}

// Example: ellipse with semi-major axis 5, semi-minor axis 3
println!("{}", ellipse_perimeter(5.0, 3.0).unwrap()); // 25.526998863398124

§Features

§Legendre’s complete integrals

  • ellipk: Complete elliptic integral of the first kind (K).
  • ellipe: Complete elliptic integral of the second kind (E).
  • ellippi: Complete elliptic integral of the third kind (Π).
  • ellipd: Complete elliptic integral of Legendre’s type (D).

§Legendre’s incomplete integrals

§Bulirsch’s integrals

  • cel: General complete elliptic integral in Bulirsch’s form.
  • cel1: Complete elliptic integral of the first kind in Bulirsch’s form.
  • cel2: Complete elliptic integral of the second kind in Bulirsch’s form.
  • el1: Incomplete elliptic integral of the first kind in Bulirsch’s form.
  • el2: Incomplete elliptic integral of the second kind in Bulirsch’s form.
  • el3: Incomplete elliptic integral of the third kind in Bulirsch’s form.

§Carlson’s symmetric integrals

  • elliprf: Symmetric elliptic integral of the first kind (RF).
  • elliprg: Symmetric elliptic integral of the second kind (RG).
  • elliprj: Symmetric elliptic integral of the third kind (RJ).
  • elliprc: Degenerate elliptic integral of RF (RC).
  • elliprd: Degenerate elliptic integral of the third kind (RD).

§Miscellaneous functions

§Feature Flags

  • unstable: Enable unstable or untested features that might be changed without notice in the future.
  • test_force_fail: Used for testing only. Force tests to reach code unreachable under normal circumstances.

§Testing

The function results are compared with Boost Math test data and Wolfram Engine test data. The accuracy report and the test data along with the test generation scripts can be found here.

§Acknowledgment

Ellip is derived from multiple mathematic libraries. We thank the opensource contributors for making mathematic libraries free for all. Following are the main original works used in the development of Ellip. Detailed credits are available in the source code.

References for original implementations are:

Unicode-style mathematical notation are created using Diagon.

§Citation

The paper describing Ellip is published in the Journal of Open Source Software. If Ellip is helpful to your work, please consider citing it:

Pornsiriprasert, S., (2026). Ellip: An Elliptic Integral Library for Rust. Journal of Open Source Software, 11(118), 9386, https://doi.org/10.21105/joss.09386

Bibtex format:

@article{Pornsiriprasert2026,
    doi = {10.21105/joss.09386},
   url = {https://doi.org/10.21105/joss.09386},
   year = {2026},
   publisher = {The Open Journal},
   volume = {11}, number = {118}, pages = {9386},
   author = {Pornsiriprasert, Sira},
   title = {Ellip: An Elliptic Integral Library for Rust},
   journal = {Journal of Open Source Software} }

Re-exports§

pub use legendre::ellipd;
pub use legendre::ellipe;
pub use legendre::ellipk;
pub use legendre::ellippi;
pub use legendre::ellipdinc;
pub use legendre::ellipeinc;
pub use legendre::ellipf;
pub use legendre::ellippiinc;
pub use legendre::ellippiinc_bulirsch;
pub use bulirsch::cel;
pub use bulirsch::cel1;
pub use bulirsch::cel2;
pub use bulirsch::el1;
pub use bulirsch::el2;
pub use bulirsch::el3;
pub use carlson::elliprc;
pub use carlson::elliprd;
pub use carlson::elliprf;
pub use carlson::elliprg;
pub use carlson::elliprj;
pub use misc::heuman_lambda;
pub use misc::jacobi_zeta;

Modules§

bulirsch
Elliptic integral functions in Bulirsch’s form.
carlson
Elliptic integral functions in Carlson’s form.
legendre
Elliptic integral functions in Legendre’s form.
misc
Functions related to elliptic integrals
util
Utility functions like assert_close.

Type Aliases§

StrErr
Static error str