NAG Library Function Document

nag_elliptic_integral_E (s21bfc)


    1  Purpose
    7  Accuracy


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


#include <nag.h>
#include <nags.h>
double  nag_elliptic_integral_E (double phi, double dm, NagError *fail)


nag_elliptic_integral_E (s21bfc) calculates an approximation to the integral
Eϕm = 0ϕ 1-m sin2θ 12 dθ ,  
where 0ϕ π2  and msin2ϕ1 .
The integral is computed using the symmetrised elliptic integrals of Carlson (Carlson (1979) and Carlson (1988)). The relevant identity is
Eϕm = sinϕ RF q,r,1 - 13 m sin3ϕ RD q,r,1 ,  
where q=cos2ϕ , r=1-m sin2ϕ , RF  is the Carlson symmetrised incomplete elliptic integral of the first kind (see nag_elliptic_integral_rf (s21bbc)) and RD  is the Carlson symmetrised incomplete elliptic integral of the second kind (see nag_elliptic_integral_rd (s21bcc)).


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


1:     phi doubleInput
2:     dm doubleInput
On entry: the arguments ϕ and m of the function.
  • 0.0phi π2;
  • dm× sin2phi 1.0 .
3:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
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.
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.
On entry, phi=value.
Constraint: 0phiπ2.
On entry, phi=value and dm=value; the integral is undefined.
Constraint: dm×sin2phi1.0.


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.

Parallelism and Performance

nag_elliptic_integral_E (s21bfc) is not threaded in any implementation.

Further Comments

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 RF  and RD , 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-ϕ|m=-Eϕ|m. A parameter m>1 can be replaced by one less than unity using Eϕ|m=mEϕm|1m-m-1ϕ.


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

Program Text

Program Text (s21bfce.c)

Program Data


Program Results

Program Results (s21bfce.r)

GnuplotProduced by GNUPLOT 5.0 patchlevel 3 1 1.1 1.2 1.3 1.4 1.5 1.6 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.8 0.9 1 1.1 1.2 1.3 Example Program Classical (Legendre) Form of the Incomplete Elliptic Integral of the Second Kind gnuplot_plot_1 f m
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017