naginterfaces.library.specfun.ellipint_​legendre_​2

naginterfaces.library.specfun.ellipint_legendre_2(phi, dm)

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_29.3/flhtml/s/s21bff.html

Parameters

phifloat

The arguments $\varphi$ and $m$ of the function.

dmfloat

The arguments $\varphi$ and $m$ of the function.

Returns

efloat

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

Raises

NagValueError

(errno $1$)

On entry, $phi=⟨value⟩$.

Constraint: $0\le phi\le \frac{\pi }{2}$.

(errno $2$)

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

Constraint: $dm\times {sin}^2\left(phi\right)\le 1.0$.

Notes

ellipint_legendre_2 calculates an approximation to the integral

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

where $0\le \varphi \le \frac{\pi }{2}$ and $m{\sin }^2\left(\varphi \right)\le 1$.

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

$$E\left(\varphi |m\right)=\sin \left(\varphi \right)R_F(q,r,1)-\frac{1}{3}m\sin \left(\varphi \right)R_D(q,r,1)\text{,}$$

where $q={\cos }^2\left(\varphi \right)$, $r=1-m{\sin }^2\left(\varphi \right)$, $R_F$ 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

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