cc [ flags ] -I/usr/local/include file(s) -L/usr/local/lib -lmcw [ ... ] #include <mathcw.h> extern float elleif (float phi, float k); extern double ellei (double phi, double k); extern long double elleil (long double phi, long double k); extern __float80 elleiw (__float80 phi, __float80 k); extern __float128 elleiq (__float128 phi, __float128 k); extern long_long_double elleill (long_long_double phi, long_long_double k); extern decimal_float elleidf (decimal_float phi, decimal_float k); extern decimal_double elleid (decimal_double phi, decimal_double k); extern decimal_long_double elleidl (decimal_long_double phi, decimal_long_double k); extern decimal_long_long_double elleidll (decimal_long_long_double phi, decimal_long_long_double k);
NB: Functions with prototypes containing underscores in type names may be available only with certain extended compilers.
ellei(phi,k) = integral(t=0:phi) (1 - (k * sin(t))**2)**(1/2) dtwhere phi is an angle in radians in [0, 2*pi] and k is in [-1,+1].
The Legendre incomplete elliptic function of the second kind in Abramowitz and Stegun's Handbook of Mathematical Functions (Chapter 17 and Table 17.6) can be computed like this:
E(phi \ alpha) = ellei(phi, sin(alpha)).
Equivalents in other systems are:
Maple: E(phi \ alpha) = EllipticE(sin(phi), sin(alpha)) Mathematica: E(phi \ alpha) = EllipticE[phi, (Sin[alpha])^2]