ELLPI 3CW "07 February 2010" "mathcw-1.00"

Table of contents


NAME

ellpif, ellpi, ellpil, ellpiw, ellpiq, ellpill, ellpidf, ellpid, ellpidl, ellpidll - incomplete elliptic integral function of the third kind

SYNOPSIS

cc [ flags ] -I/usr/local/include file(s) -L/usr/local/lib -lmcw [ ... ]

#include <mathcw.h>

extern float ellpif (float n, float phi, float k);

extern double ellpi (double n, double phi, double k);

extern long double ellpil (long double n, long double phi, long double k);

extern __float80 ellpiw (__float80 n, __float80 phi, __float80 k);

extern __float128 ellpiq (__float128 n, __float128 phi, __float128 k);

extern long_long_double ellpill (long_long_double n, long_long_double phi, long_long_double k);

extern decimal_float ellpidf (decimal_float n, decimal_float phi, decimal_float k);

extern decimal_double ellpid (decimal_double n, decimal_double phi, decimal_double k);

extern decimal_long_double ellpidl (decimal_long_double n, decimal_long_double phi, decimal_long_double k);

extern decimal_long_long_double ellpidll (decimal_long_long_double n, 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.


DESCRIPTION

Compute the incomplete elliptic integral function of the third kind defined by
ellpi(n,phi,k) = integral(t=0:phi) (1 - n * (sin(t))**2)**(-1) * (1 - (k * sin(t))**2)**(-1/2) dt
where n is a nonnegative real number, phi is an angle in radians in [0, 2*pi], and k is in [-1,+1].

The Legendre incomplete elliptic function of the third kind in Abramowitz and Stegun's Handbook of Mathematical Functions (Chapter 17 and Table 17.9) can be computed like this:

\(*P(n; phi \ alpha) = ellpi(n, phi, sin(alpha)).

Equivalents in other systems are:

Maple: \(*P(n; phi \ alpha) = EllipticPi(sin(phi), n, sin(alpha)) Mathematica: \(*P(n; phi \ alpha) = EllipticPi[n, phi, (Sin[alpha])^2]

RETURN VALUES

Return the value of the incomplete elliptic integral function of the third kind.

ERRORS

If any argument is a NaN, return that argument and set errno to EDOM. If the arguments n or k are out of the range [-1,+1], return a NaN and set errno to EDOM.

SEE ALSO

agm(3CW), elk(3CW), elkm1(3CW), elldi(3CW), elle(3CW), ellec(3CW), ellei(3CW), ellfi(3CW), ellk(3CW), ellkc(3CW), ellrc(3CW), ellrd(3CW), ellre(3CW), ellrf(3CW), ellrg(3CW), ellrh(3CW), ellrj(3CW), ellrk(3CW), ellrl(3CW), ellrm(3CW), elq(3CW), elq1p(3CW), elqc(3CW), elqc1p(3CW).