# naginterfaces.library.specfun.ellipint_​legendre_​1¶

naginterfaces.library.specfun.ellipint_legendre_1(phi, dm)[source]

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_30.1/flhtml/s/s21bef.html

Parameters
phifloat

The arguments and of the function.

dmfloat

The arguments and of the function.

Returns
ffloat

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

Raises
NagValueError
(errno )

On entry, .

Constraint: .

On failure, the function returns zero.

(errno )

On entry, and ; the integral is undefined.

Constraint: .

On failure, the function returns zero.

Warns
NagAlgorithmicWarning
(errno )

On entry, and ; 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

where , and and 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

where , and 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