naginterfaces.library.specfun.ellipint_​general_​2

naginterfaces.library.specfun.ellipint_general_2(z, akp, a, b)[source]

ellipint_general_2 returns the value of the general elliptic integral of the second kind for a complex argument .

For full information please refer to the NAG Library document for s21da

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s21daf.html

Parameters
zcomplex

The argument of the function.

akpfloat

The argument of the function.

afloat

The argument of the function.

bfloat

The argument of the function.

Returns
fcomplex

The value of the general elliptic integral of the second kind for a complex argument .

Raises
NagValueError
(errno )

On entry, is too large: . It must not exceed .

(errno )

On entry, is too large: . It must not exceed .

(errno )

On entry, : .

(errno )

On entry, is too large: . It must not exceed .

(errno )

The iterative procedure used to evaluate the integral has failed to converge.

Notes

ellipint_general_2 evaluates an approximation to the general elliptic integral of the second kind given by

where and are real arguments, is a complex argument whose real part is non-negative and is a real argument (the complementary modulus). The evaluation of is based on the Gauss transformation. Further details, in particular for the conformal mapping provided by , can be found in Bulirsch (1960).

Special values include

or (the elliptic integral of the first kind) and

or (the elliptic integral of the second kind). Note that the values of and are equal to in the trivial case .

ellipint_general_2 is derived from an Algol 60 procedure given by Bulirsch (1960). Constraints are placed on the values of and in order to avoid the possibility of machine overflow.

References

Bulirsch, R, 1960, Numerical calculation of elliptic integrals and elliptic functions, Numer. Math. (7), 76–90