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:     dndoubleInput
2:     phidoubleInput
3:     dmdoubleInput
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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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.
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

Not applicable.

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)