naginterfaces.library.specfun.ellipint_​legendre_​1

naginterfaces.library.specfun.ellipint_legendre_1(phi, dm)

ellipint_legendre_1 returns a value of the classical (Legendre) form of the incomplete elliptic integral of the first kind.

For full information please refer to the NAG Library document for s21be

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/s/s21bef.html

Parameters

phifloat

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

dmfloat

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

Returns

ffloat

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

Raises

NagValueError

(errno $1$)

On entry, $phi=⟨value⟩$.

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

On failure, the function returns zero.

(errno $2$)

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

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

On failure, the function returns zero.

Warns

NagAlgorithmicWarning

(errno $3$)

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

On failure, the function returns the largest machine number (see machine.real_largest).

Notes

ellipint_legendre_1 calculates an approximation to the integral

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

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

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

$$F\left(\varphi |m\right)=R_F(q,r,1)\sin \left(\varphi \right)\text{,}$$

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

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