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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_ellipint_legendre_2 (s21bf)

## Purpose

nag_specfun_ellipint_legendre_2 (s21bf) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind, via the function name.

## Syntax

[result, ifail] = s21bf(phi, dm)
[result, ifail] = nag_specfun_ellipint_legendre_2(phi, dm)

## Description

nag_specfun_ellipint_legendre_2 (s21bf) calculates an approximation to the integral
 $Eϕ∣m = ∫0ϕ 1-m sin2⁡θ 12 dθ ,$
where $0\le \varphi \le \frac{\pi }{2}$ and $m{\mathrm{sin}}^{2}\varphi \le 1$.
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
 $Eϕ∣m = sin⁡ϕ RF q,r,1 - 13 m sin3⁡ϕ RD q,r,1 ,$
where $q={\mathrm{cos}}^{2}\varphi$, $r=1-m{\mathrm{sin}}^{2}\varphi$, ${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_specfun_ellipint_symm_1 (s21bb)) and ${R}_{D}$ is the Carlson symmetrised incomplete elliptic integral of the second kind (see nag_specfun_ellipint_symm_2 (s21bc)).

## 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:     $\mathrm{phi}$ – double scalar
2:     $\mathrm{dm}$ – double scalar
The arguments $\varphi$ and $m$ of the function.
Constraints:
• $0.0\le {\mathbf{phi}}\le \frac{\pi }{2}$;
• ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.

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:
${\mathbf{ifail}}=1$
Constraint: $0\le {\mathbf{phi}}\le \frac{\pi }{2}$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
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_2 (s21bf) 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 ${R}_{D}$, see the function documents for nag_specfun_ellipint_symm_1 (s21bb) and nag_specfun_ellipint_symm_2 (s21bc), 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. For example, $E\left(-\varphi |m\right)=-E\left(\varphi |m\right)$. A parameter $m>1$ can be replaced by one less than unity using $E\left(\varphi |m\right)=\sqrt{m}E\left(\varphi \sqrt{m}|\frac{1}{m}\right)-\left(m-1\right)\varphi$.

## Example

This example simply generates a small set of nonextreme arguments that are used with the function to produce the table of results.
```function s21bf_example

fprintf('s21bf example results\n\n');

phi = [pi/6  pi/3   pi/2];
dm  = [1/4   1/2    3/4];
result = phi;

for j = 1:numel(phi)
[result(j), ifail] = s21bf(phi(j), dm(j));
end

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

s21bf_plot;

function s21bf_plot
phi = [1:0.02:1.56];
m = [0.5:0.02:0.98,0.982:0.002:1];
F = zeros(numel(phi),numel(m));
for i = 1:numel(phi)
for j = 1:numel(m)
[F(i,j), ifail] = s21bf(phi(i), m(j));
end
end

fig1 = figure;
[Y,X] = meshgrid(m,phi);
surf(X,Y,F);
xlabel('\Phi');
ylabel('m');
zlabel('F(\Phi,n)');
title({'Incomplete Elliptic Integral of the second kind', ...
'Classical (Legendre) form'});

```
```s21bf example results

phi      m        E(phi|m)
0.52    0.25       0.5179
1.05    0.50       0.9650
1.57    0.75       1.2111
``` 