naginterfaces.library.specfun.ellipint_​legendre_​2

naginterfaces.library.specfun.ellipint_legendre_2(phi, dm)[source]

ellipint_legendre_2 returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind.

For full information please refer to the NAG Library document for s21bf

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s21bff.html

Parameters
phifloat

The arguments and of the function.

dmfloat

The arguments and of the function.

Returns
efloat

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

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, and ; the integral is undefined.

Constraint: .

Notes

ellipint_legendre_2 calculates an approximation to the integral

where 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 second kind (see ellipint_symm_2()).

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