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ellipj

Compute the Jacobi elliptic functions sn, cn, and dn.

The Jacobi elliptic functions may be defined as the inverse of the incomplete elliptic integral of the first kind. Accordingly, they compute the value φ which satisfies the equation

where the parameter m is related to the modulus k by m = k^2.

Installation

npm install @stdlib/math-base-special-ellipj

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var ellipj = require( '@stdlib/math-base-special-ellipj' );

ellipj( u, m )

Computes the Jacobi elliptic functions functions sn, cn, and dn, and the Jacobi amplitude am.

var v = ellipj( 0.3, 0.5 );
// returns [ ~0.293, ~0.956, ~0.978, ~0.298 ]

v = ellipj( 0.0, 0.0 );
// returns [ ~0.0, ~1.0, ~1.0, ~0.0 ]

v = ellipj( Infinity, 1.0 );
// returns [ ~1.0, ~0.0, ~0.0, ~1.571 ]

v = ellipj( 0.0, -2.0 );
// returns [ ~0.0, ~1.0, ~1.0, NaN ]

v = ellipj( NaN, NaN );
// returns [ NaN, NaN, NaN, NaN ]

ellipj.assign( u, m, out, stride, offset )

Computes the Jacobi elliptic functions sn, cn, dn, and Jacobi amplitude am and assigns results to a provided output array.

var Float64Array = require( '@stdlib/array-float64' );

var out = new Float64Array( 4 );

var v = ellipj.assign( 0.0, 0.0, out, 1, 0 );
// returns <Float64Array>[ ~0.0, ~1.0, ~1.0, ~0.0 ]

var bool = ( v === out );
// returns true

ellipj.sn( u, m )

Computes the Jacobi elliptic function sn of value u with modulus m.

var v = ellipj.sn( 0.3, 0.5 );
// returns ~0.293

ellipj.cn( u, m )

Computes the Jacobi elliptic function cn of value u with modulus m.

var v = ellipj.cn( 0.3, 0.5 );
// returns ~0.956

ellipj.dn( u, m )

Computes the Jacobi elliptic function dn of value u with modulus m.

var v = ellipj.dn( 0.3, 0.5 );
// returns ~0.978

ellipj.am( u, m )

Computes the Jacobi amplitude am of value u with modulus m.

var v = ellipj.am( 0.3, 0.5 );
// returns ~0.298

v = ellipj.am( 0.3, 2.0 );
// returns NaN

Although sn, cn, and dn may be computed for -∞ < m < ∞, the domain of am is 0 ≤ m ≤ 1. For m < 0 or m > 1, the function returns NaN.

Notes

• Functions sn, cn, and dn are valid for -∞ < m < ∞. Values for m < 0 or m > 1 are computed in terms of Jacobi elliptic functions with 0 < m < 1 via the transformations outlined in Equations 16.13 and 16.15 from The Handbook of Mathematical Functions (Abramowitz and Stegun).
• If more than one of sn, cn, dn, or am is to be computed, preferring using ellipj to compute all four values simultaneously.

Examples

var linspace = require( '@stdlib/array-base-linspace' );
var ellipk = require( '@stdlib/math-base-special-ellipk' );
var ellipj = require( '@stdlib/math-base-special-ellipj' );

var m = 0.7;
var u = linspace( 0.0, ellipk( m ), 100 );

var out;
var i;
for ( i = 0; i < 100; i++ ) {
    out = ellipj( u[ i ], m );
    console.log( 'sn(%d, %d) = %d', u[ i ], m, out[ 0 ] );
    console.log( 'cn(%d, %d) = %d', u[ i ], m, out[ 1 ] );
    console.log( 'dn(%d, %d) = %d', u[ i ], m, out[ 2 ] );
    console.log( 'am(%d, %d) = %d', u[ i ], m, out[ 3 ] );
}

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References

• Fukushima, Toshio. 2009. "Fast computation of complete elliptic integrals and Jacobian elliptic functions." Celestial Mechanics and Dynamical Astronomy 105 (4): 305. doi:10.1007/s10569-009-9228-z.
• Fukushima, Toshio. 2015. "Precise and fast computation of complete elliptic integrals by piecewise minimax rational function approximation." Journal of Computational and Applied Mathematics 282 (July): 71–76. doi:10.1016/j.cam.2014.12.038.

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See Also

• @stdlib/math-base/special/ellipe: compute the complete elliptic integral of the second kind.
• @stdlib/math-base/special/ellipk: compute the complete elliptic integral of the first kind.

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.