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.

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 ] );
}

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.