Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind

Much simplified expressions for certain complete elliptic integralsin terms of the beta function are produced. The reverse procedureof expressing gamma functions in terms of complete ellipticintegrals allows us to use the arithmetic-geometric mean iterationto compute gamma functions, using quadratically convergent iterations.Explicit algorithms are given for computing (n/4) and (n/6),where 1m3 and 1n5.