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 integrals in terms of the beta function are produced. The reverse procedure of expressing gamma functions in terms of complete elliptic integrals allows us to use the arithmetic- geometric mean iteration to compute gamma functions, using quadratically convergent iterations. Explicit algorithms are given for computing Γ(m/4) and Γ(n/6), where 1≤m≤3 and 1≤n≤5