ellpk

Complete elliptic integral of the first kind.

SYNOPSIS:

double m1, y, ellpk();
y = ellpk(m1);

DESCRIPTION:

Approximates the integral:

            pi/2
             -
            | |
            |           dt
K(m) =     |    ------------------
            |                  2
          | |    sqrt(1 - m sin t)
           -
            0

where m = 1 - m1, using the approximation:

     P(x) - log x Q(x).

The argument m1 is used rather than m, so that the logarithmic singularity at m = 1 will be shifted to the origin; this preserves maximum accuracy.

K(0) = pi/2.

ACCURACY:

                      Relative error:
arithmetic   domain     # trials      peak         rms
    DEC        0,1        16000       3.5e-17     1.1e-17
    IEEE       0,1        30000       2.5e-16     6.8e-17

ERROR MESSAGES:

message       condition      value returned
domain        x<0, x>1           0.0