Carlson R_j

(carlson-rj x y z p)

Carlson's R_j is defined by

 Rj(x,y,z,p) = integrate(3/2*(t+x)^(-1/2)*(t+y)^(-1/2)*(t+z)^(-1/2)*(t+p)^(-1), t, 0, inf)

It is related to the elliptic integral of the third kind by

  P(phi,k,n) = integrate((1+n*sin(t)^2)^(-1)*(1-k^2*sin(t)^2)^(-1/2), t, 0, phi)
             = sin(phi)*Rf(cos(phi)^2, 1-k^2*sin(phi)^2, 1)
                - (n/3)*sin(phi)^3*Rj(cos(phi)^2, 1-k^2*sin(phi)^2, 1, 1+n*sin(phi)^2)

Note that Carlson's definition of the integrals has the opposite sign for the parameter n compared to the definition given by A&S.