Carlson R_f

(carlson-rf x y z)

Carlson's R_f function is defined by

Rf(x, y, z) = 1/2*integrate((t+x)^(-1/2)*(t+y)^(-1/2)*(t+z)^(-1/2), t, 0, inf)

This is related to the incomplete elliptic integral of the second kind by

 E(phi, m) = sin(phi) * Rf(cos(phi)^2, 1 - m*sin(phi)^2, 1)
              - (m/3)*sin(phi)^3*Rd(cos(phi)^2, 1 - m*sin(phi)^2, 1)

Obviously, the complete integral is

 E(m) = Rf(0, 1 - m, 1) - (m/3)*Rd(0, 1 - m, 1)