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# NAG Toolbox: nag_specfun_ellipint_legendre_3 (s21bg)

## Purpose

nag_specfun_ellipint_legendre_3 (s21bg) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind, via the function name.

## Syntax

[result, ifail] = s21bg(dn, phi, dm)
[result, ifail] = nag_specfun_ellipint_legendre_3(dn, phi, dm)

## Description

nag_specfun_ellipint_legendre_3 (s21bg) calculates an approximation to the integral
 $Π n;ϕ∣m = ∫0ϕ 1-n sin2⁡θ -1 1-m sin2⁡θ -12 dθ ,$
where $0\le \varphi \le \frac{\pi }{2}$, $m{\mathrm{sin}}^{2}\varphi \le 1$, $m$ and $\mathrm{sin}\varphi$ may not both equal one, and $n{\mathrm{sin}}^{2}\varphi \ne 1$.
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
 $Π n;ϕ∣m = sin⁡ϕ RF q,r,1 + 13 n sin3⁡ϕ RJ q,r,1,s ,$
where $q={\mathrm{cos}}^{2}\varphi$, $r=1-m{\mathrm{sin}}^{2}\varphi$, $s=1-n{\mathrm{sin}}^{2}\varphi$, ${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_specfun_ellipint_symm_1 (s21bb)) and ${R}_{J}$ is the Carlson symmetrised incomplete elliptic integral of the third kind (see nag_specfun_ellipint_symm_3 (s21bd)).

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{dn}$ – double scalar
2:     $\mathrm{phi}$ – double scalar
3:     $\mathrm{dm}$ – double scalar
The arguments $n$, $\varphi$ and $m$ of the function.
Constraints:
• $0.0\le {\mathbf{phi}}\le \frac{\pi }{2}$;
• ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$;
• Only one of $\mathrm{sin}\left({\mathbf{phi}}\right)$ and dm may be $1.0$;
• ${\mathbf{dn}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\ne 1.0$.
Note that ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)=1.0$ is allowable, as long as ${\mathbf{dm}}\ne 1.0$.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
Constraint: $0\le {\mathbf{phi}}\le \left(\pi /2\right)$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
W  ${\mathbf{ifail}}=3$
On entry, $\mathrm{sin}\left({\mathbf{phi}}\right)=1$ and ${\mathbf{dm}}=1.0$; the integral is infinite.
W  ${\mathbf{ifail}}=4$
Constraint: ${\mathbf{dn}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\ne 1.0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

In principle nag_specfun_ellipint_legendre_3 (s21bg) is capable of producing full machine precision. However round-off errors in internal arithmetic will result in slight loss of accuracy. This loss should never be excessive as the algorithm does not involve any significant amplification of round-off error. It is reasonable to assume that the result is accurate to within a small multiple of the machine precision.

You should consult the S Chapter Introduction, which shows the relationship between this function and the Carlson definitions of the elliptic integrals. In particular, the relationship between the argument-constraints for both forms becomes clear.
For more information on the algorithms used to compute ${R}_{F}$ and ${R}_{J}$, see the function documents for nag_specfun_ellipint_symm_1 (s21bb) and nag_specfun_ellipint_symm_3 (s21bd), respectively.
If you wish to input a value of phi outside the range allowed by this function you should refer to Section 17.4 of Abramowitz and Stegun (1972) for useful identities.

## Example

This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
```function s21bg_example

fprintf('s21bg example results\n\n');

dn  = [0.1   -0.2   0.3];
phi = [pi/6  pi/3   pi/2];
dm  = [1/4   1/2    3/4];
result = phi;

for j = 1:numel(phi)
[result(j), ifail] = s21bg(dn(j), phi(j), dm(j));
end

fprintf('    n       phi      m       Pi(n;phi|m)\n');
fprintf(' %7.2f %7.2f %7.2f %12.4f\n', [dn; phi; dm; result]);

```
```s21bg example results

n       phi      m       Pi(n;phi|m)
0.10    0.52    0.25       0.5341
-0.20    1.05    0.50       1.0778
0.30    1.57    0.75       2.6568
```

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