# naginterfaces.library.specfun.ellipint_​symm_​3¶

naginterfaces.library.specfun.ellipint_symm_3(x, y, z, r)[source]

ellipint_symm_3 returns a value of the symmetrised elliptic integral of the third kind.

For full information please refer to the NAG Library document for s21bd

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s21bdf.html

Parameters
xfloat

The arguments , , and of the function.

yfloat

The arguments , , and of the function.

zfloat

The arguments , , and of the function.

rfloat

The arguments , , and of the function.

Returns
rjfloat

The value of the symmetrised elliptic integral of the third kind.

Raises
NagValueError
(errno )

On entry, , and .

Constraint: at most one of , and is .

The function is undefined.

(errno )

On entry, , and .

Constraint: and and .

(errno )

On entry, .

Constraint: .

(errno )

On entry, , , , and .

Constraint: and at most one of , and is less than .

(errno )

On entry, , , , and .

Constraint: and and and .

Notes

ellipint_symm_3 calculates an approximation to the integral

where , , , and at most one of , and is zero.

If , the result computed is the Cauchy principal value of the integral.

The basic algorithm, which is due to Carlson (1979) and Carlson (1988), is to reduce the arguments recursively towards their mean by the rule:

For sufficiently large,

and the function may be approximated by a fifth order power series

where .

The truncation error in this expansion is bounded by and the recursion process is terminated when this quantity is negligible compared with the machine precision. The function may fail either because it has been called with arguments outside the domain of definition or with arguments so extreme that there is an unavoidable danger of setting underflow or overflow.

Note: , so there exists a region of extreme arguments for which the function value is not representable.

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