s21bf returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind.

# Syntax

C#
```public static double s21bf(
double phi,
double dm,
out int ifail
)```
Visual Basic
```Public Shared Function s21bf ( _
phi As Double, _
dm As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s21bf(
double phi,
double dm,
[OutAttribute] int% ifail
)```
F#
```static member s21bf :
phi : float *
dm : float *
ifail : int byref -> float
```

#### Parameters

phi
Type: System..::..Double
On entry: the arguments $\varphi$ and $m$ of the function.
Constraints:
• $0.0\le {\mathbf{phi}}\le \frac{\pi }{2}$;
• ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
dm
Type: System..::..Double
On entry: the arguments $\varphi$ and $m$ of the function.
Constraints:
• $0.0\le {\mathbf{phi}}\le \frac{\pi }{2}$;
• ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
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

s21bf returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind.

# Description

s21bf calculates an approximation to the integral
 $Eϕ∣m=∫0ϕ1-msin2 θ12dθ,$
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
 $Eϕ∣m=sin ϕRFq,r,1-13msin3 ϕRDq,r,1,$
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 s21bb) and ${R}_{D}$ is the Carlson symmetrised incomplete elliptic integral of the second kind (see 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

# 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.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# Accuracy

In principle 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.

# 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 ${R}_{D}$, see the method documents for s21bb and s21bc, 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. 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 method to produce the table of results.

Example program (C#): s21bfe.cs

Example program results: s21bfe.r