# naginterfaces.library.specfun.ellipint_​legendre_​3¶

naginterfaces.library.specfun.ellipint_legendre_3(dn, phi, dm)[source]

ellipint_legendre_3 returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind.

For full information please refer to the NAG Library document for s21bg

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s21bgf.html

Parameters
dnfloat

The arguments , and of the function.

phifloat

The arguments , and of the function.

dmfloat

The arguments , and of the function.

Returns
pfloat

The value of the classical (Legendre) form of the incomplete elliptic integral of the third kind.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, and ; the integral is undefined.

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

On entry, and ; the integral is infinite.

(errno )

On entry, and ; the integral is infinite.

Constraint: .

Notes

ellipint_legendre_3 calculates an approximation to the integral

where , , and may not both equal one, and .

The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is

where , , , is the Carlson symmetrised incomplete elliptic integral of the first kind (see ellipint_symm_1()) and is the Carlson symmetrised incomplete elliptic integral of the third kind (see ellipint_symm_3()).

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Carlson, B C, 1979, Computing elliptic integrals by duplication, Numerische Mathematik (33), 1–16

Carlson, B C, 1988, A table of elliptic integrals of the third kind, Math. Comput. (51), 267–280