s21bh returns a value of the classical (Legendre) form of the complete elliptic integral of the first kind.

# Syntax

C# |
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public static double s21bh( double dm, out int ifail ) |

Visual Basic |
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Public Shared Function s21bh ( _ dm As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double |

Visual C++ |
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public: static double s21bh( double dm, [OutAttribute] int% ifail ) |

F# |
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static member s21bh : dm : float * ifail : int byref -> float |

#### Parameters

- dm
- Type: System..::..Double
*On entry*: the argument $m$ of the function.*Constraint*: ${\mathbf{dm}}<1.0$.

- 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

s21bh returns a value of the classical (Legendre) form of the complete elliptic integral of the first kind.

# Description

s21bh calculates an approximation to the integral

where $m<1$.

$$K\left(m\right)=\underset{0}{\overset{\frac{\pi}{2}}{\int}}{\left(1-m{\mathrm{sin}}^{2}\u200a\theta \right)}^{-\frac{1}{2}}d\theta \text{,}$$ |

The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is

where ${R}_{F}$ is the Carlson symmetrised incomplete elliptic integral of the first kind (see s21bb).

$$K\left(m\right)={R}_{F}\left(0,1-m,1\right)\text{,}$$ |

# References

Abramowitz M and Stegun I A (1972)

*Handbook of Mathematical Functions*(3rd Edition) Dover PublicationsCarlson B C (1979) Computing elliptic integrals by duplication

*Numerische Mathematik***33**1–16Carlson 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:

- ${\mathbf{ifail}}=1$
- On entry, ${\mathbf{dm}}>1.0$; the function is undefined. On failure, the method returns zero.

- ${\mathbf{ifail}}=2$
- On entry, ${\mathbf{dm}}=1.0$; the function is infinite. On failure, the method returns the largest machine number (see x02al).

# Accuracy

In principle s21bh 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.

# Further Comments

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 algorithm used to compute ${R}_{F}$, see the method document for s21bb.

# Example

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