naginterfaces.library.specfun.ellipint_​complete_​2

naginterfaces.library.specfun.ellipint_complete_2(dm)

ellipint_complete_2 returns a value of the classical (Legendre) form of the complete elliptic integral of the second kind.

For full information please refer to the NAG Library document for s21bj

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s21bjf.html

Parameters

dmfloat

The argument $m$ of the function.

Returns

efloat

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

Raises

NagValueError

(errno $1$)

On entry, $dm=⟨value⟩$; the integral is undefined.

Constraint: $dm\le 1.0$.

Notes

ellipint_complete_2 calculates an approximation to the integral

$$E\left(m\right)={\int }_0^{\frac{\pi }{2}}{(1-m{\sin }^2\left(\theta \right))}^{\frac{1}{2}}d\theta \text{,}$$

where $m\le 1$.

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

$$E\left(m\right)=R_F(0,1-m,1)-\frac{1}{3}m{R}_D(0,1-m,1)\text{,}$$

where $R_F$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see ellipint_symm_1()) and $R_D$ is the Carlson symmetrised incomplete elliptic integral of the second kind (see ellipint_symm_2()).

References

NIST Digital Library of Mathematical Functions

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