naginterfaces.library.specfun.ellipint_​legendre_​3

naginterfaces.library.specfun.ellipint_legendre_3(dn, phi, dm)

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://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s21bgf.html

Parameters

dnfloat

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

phifloat

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

dmfloat

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

Returns

pfloat

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

Raises

NagValueError

(errno $1$)

On entry, $phi=⟨value⟩$.

Constraint: $0\le phi\le \left(\pi /2\right)$.

(errno $2$)

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

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

Warns

NagAlgorithmicWarning

(errno $3$)

On entry, $\sin \left(phi\right)=1$ and $dm=1.0$; the integral is infinite.

(errno $4$)

On entry, $phi=⟨value⟩$ and $dn=⟨value⟩$; the integral is infinite.

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

Notes

ellipint_legendre_3 calculates an approximation to the integral

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

where $0\le \varphi \le \frac{\pi }{2}$, $m{\sin }^2\left(\varphi \right)\le 1$, $m$ and $\sin \left(\varphi \right)$ may not both equal one, and $n{\sin }^2\left(\varphi \right)\ne 1$.

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

$$\Pi (n;\varphi |m)=\sin \left(\varphi \right)R_F(q,r,1)+\frac{1}{3}n\sin \left(\varphi \right)R_J(q,r,1,s)\text{,}$$

where $q={\cos }^2\left(\varphi \right)$, $r=1-m{\sin }^2\left(\varphi \right)$, $s=1-n{\sin }^2\left(\varphi \right)$, $R_F$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see ellipint_symm_1()) and $R_J$ 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