﻿ s21bg Method
s21bg returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind.

# Syntax

C#
```public static double s21bg(
double dn,
double phi,
double dm,
out int ifail
)```
Visual Basic
```Public Shared Function s21bg ( _
dn As Double, _
phi As Double, _
dm As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s21bg(
double dn,
double phi,
double dm,
[OutAttribute] int% ifail
)```
F#
```static member s21bg :
dn : float *
phi : float *
dm : float *
ifail : int byref -> float
```

#### Parameters

dn
Type: System..::..Double
On entry: 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 ${\mathbf{dm}}\ne 1.0$.
phi
Type: System..::..Double
On entry: 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 ${\mathbf{dm}}\ne 1.0$.
dm
Type: System..::..Double
On entry: 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 ${\mathbf{dm}}\ne 1.0$.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s21bg returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind.

# Description

s21bg calculates an approximation to the integral
 $Πn;ϕ∣m=∫0ϕ1-nsin2 θ-11-msin2 θ-12dθ,$
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
 $Πn;ϕ∣m=sin ϕRFq,r,1+13nsin3 ϕRJq,r,1,s,$
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 s21bb) and ${R}_{J}$ is the Carlson symmetrised incomplete elliptic integral of the third kind (see 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

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
phi lies outside the range $\left[0,\frac{\pi }{2}\right]$. On failure, the method returns zero.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)>1.0$; the function is undefined. On failure, the method returns zero.
${\mathbf{ifail}}=3$
On entry, $\mathrm{sin}\left({\mathbf{phi}}\right)=1.0$ and ${\mathbf{dm}}=1.0$; the function is infinite. On failure, the method returns the largest machine number (see x02al).
${\mathbf{ifail}}=4$
On entry, the product ${\mathbf{dn}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)=1.0$; the function is infinite. On failure, the method returns the largest machine number (see x02al).
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

In principle 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.

# Parallelism and Performance

None.

You should consult (S not in this release), which shows the relationship between this method 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 method documents for s21bb and s21bd, respectively.
If you wish to input a value of phi outside the range allowed by this method 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 method to produce the table of results.

Example program (C#): s21bge.cs

Example program results: s21bge.r