s Chapter Contents
s Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_elliptic_integral_pi (s21bgc)

## 1  Purpose

nag_elliptic_integral_pi (s21bgc) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the third kind.

## 2  Specification

 #include #include
 double nag_elliptic_integral_pi (double dn, double phi, double dm, NagError *fail)

## 3  Description

nag_elliptic_integral_pi (s21bgc) 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_elliptic_integral_rf (s21bbc)) and ${R}_{J}$ is the Carlson symmetrised incomplete elliptic integral of the third kind (see nag_elliptic_integral_rj (s21bdc)).

## 4  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

## 5  Arguments

1:    $\mathbf{dn}$doubleInput
2:    $\mathbf{phi}$doubleInput
3:    $\mathbf{dm}$doubleInput
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 ${\mathbf{dm}}\ne 1.0$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL
On entry, ${\mathbf{phi}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{phi}}\le \left(\pi /2\right)$.
NE_REAL_2
On entry, ${\mathbf{phi}}=〈\mathit{\text{value}}〉$ and ${\mathbf{dm}}=〈\mathit{\text{value}}〉$; the integral is undefined.
Constraint: ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
On entry, ${\mathbf{phi}}=〈\mathit{\text{value}}〉$ and ${\mathbf{dn}}=〈\mathit{\text{value}}〉$; the integral is infinite.
Constraint: ${\mathbf{dn}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\ne 1.0$.
NW_INTEGRAL_INFINITE
On entry, $\mathrm{sin}\left({\mathbf{phi}}\right)=1$ and ${\mathbf{dm}}=1.0$; the integral is infinite.

## 7  Accuracy

In principle nag_elliptic_integral_pi (s21bgc) 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.

## 8  Parallelism and Performance

nag_elliptic_integral_pi (s21bgc) is not threaded in any implementation.

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_elliptic_integral_rf (s21bbc) and nag_elliptic_integral_rj (s21bdc), 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.

## 10  Example

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

### 10.1  Program Text

Program Text (s21bgce.c)

None.

### 10.3  Program Results

Program Results (s21bgce.r)