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

naginterfaces.library.specfun.ellipint_complete_1(dm)[source]

ellipint_complete_1 returns a value of the classical (Legendre) form of the complete elliptic integral of the first kind.

For full information please refer to the NAG Library document for s21bh

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s21bhf.html

Parameters
dmfloat

The argument of the function.

Returns
kfloat

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

Raises
NagValueError
(errno )

On entry, ; the integral is undefined.

Constraint: .

On failure, the function returns zero.

Warns
NagAlgorithmicWarning
(errno )

On entry, ; the integral is infinite.

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

Notes

ellipint_complete_1 calculates an approximation to the integral

where .

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

where is the Carlson symmetrised incomplete elliptic integral of the first kind (see ellipint_symm_1()).

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