| 1 | = Carlson ''R'',,''j'',, = |

| 2 | |

| 3 | {{{(carlson-rj x y z p)}}} |

| 4 | |

| 5 | Carlson's ''R'',,''j'',, is defined by |

| 6 | {{{ |

| 7 | 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) |

| 8 | }}} |

| 9 | |

| 10 | It is related to the elliptic integral of the third kind by |

| 11 | {{{ |

| 12 | P(phi,k,n) = integrate((1+n*sin(t)^2)^(-1)*(1-k^2*sin(t)^2)^(-1/2), t, 0, phi) |

| 13 | = sin(phi)*Rf(cos(phi)^2, 1-k^2*sin(phi)^2, 1) |

| 14 | - (n/3)*sin(phi)^3*Rj(cos(phi)^2, 1-k^2*sin(phi)^2, 1, 1+n*sin(phi)^2) |

| 15 | }}} |

| 16 | |

| 17 | Note that Carlson's definition of the integrals has the opposite sign |

| 18 | for the parameter ''n'' compared to the definition given by A&S. |

| 19 | |