Procedures

Compute an approximation of the Carlson elliptic integral: $$R_C(x,y) = \frac{1}{2} \int_{0}^{\infty} (t+x)^{-1/2}(t+y)^{-1} dt$$ where $x\ge0$ and $y>0$.

Compute an approximation for the incomplete or complete elliptic integral of the 2nd kind: $$R_D(x,y,z) = \frac{3}{2} \int_{0}^{\infty} (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-3/2} dt$$ Where $x\ge0$, $y\ge0$, $x+y>0$, and $z>0$.

Compute an approximation for the incomplete or complete elliptic integral of the 1st kind: $$R_F(x,y,z) = \frac{1}{2} \int_{0}^{\infty} (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} dt$$ Where $x\ge0$, $y\ge0$, $z\ge0$, and at most one of them is $=0$.

Compute an approximation for the incomplete or complete elliptic integral of the 3rd kind: $$R_J(x,y,z,p) = \frac{3}{2} \int_{0}^{\infty} (t+x)^{-1/2} (t+y)^{-1/2} (t+z)^{-1/2} (t+p)^{-1} dt$$ where $x\ge0$, $y\ge0$, and $z\ge0$, and at most one of them $=0$, and $p>0$.