# NAG C Library Function Document

## 1Purpose

nag_elliptic_integral_E (s21bfc) returns a value of the classical (Legendre) form of the incomplete elliptic integral of the second kind.

## 2Specification

 #include #include
 double nag_elliptic_integral_E (double phi, double dm, NagError *fail)

## 3Description

nag_elliptic_integral_E (s21bfc) calculates an approximation to the integral
 $Eϕ∣m = ∫0ϕ 1-m sin2⁡θ 12 dθ ,$
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
 $Eϕ∣m = sin⁡ϕ RF q,r,1 - 13 m sin3⁡ϕ RD q,r,1 ,$
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 nag_elliptic_integral_rf (s21bbc)) and ${R}_{D}$ is the Carlson symmetrised incomplete elliptic integral of the second kind (see nag_elliptic_integral_rd (s21bcc)).

## 4References

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

## 5Arguments

1:    $\mathbf{phi}$doubleInput
2:    $\mathbf{dm}$doubleInput
On entry: the arguments $\varphi$ and $m$ of the function.
Constraints:
• $0.0\le {\mathbf{phi}}\le \frac{\pi }{2}$;
• ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.
3:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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: $0\le {\mathbf{phi}}\le \frac{\pi }{2}$.
NE_REAL_2
On entry, ${\mathbf{phi}}=〈\mathit{\text{value}}〉$ and ${\mathbf{dm}}=〈\mathit{\text{value}}〉$; the integral is undefined.
Constraint: ${\mathbf{dm}}×{\mathrm{sin}}^{2}\left({\mathbf{phi}}\right)\le 1.0$.

## 7Accuracy

In principle nag_elliptic_integral_E (s21bfc) 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.

## 8Parallelism and Performance

nag_elliptic_integral_E (s21bfc) 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 ${R}_{D}$, see the function documents for nag_elliptic_integral_rf (s21bbc) and nag_elliptic_integral_rd (s21bcc), 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. 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 $E\left(\varphi |m\right)=\sqrt{m}E\left(\varphi \sqrt{m}|\frac{1}{m}\right)-\left(m-1\right)\varphi$.

## 10Example

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

### 10.1Program Text

Program Text (s21bfce.c)

None.

### 10.3Program Results

Program Results (s21bfce.r)