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

1992/01/01 by Jonathan M. Borwein, I. J. Zucker · 2 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Applied mathematics #Electromagnetic Scattering and Analysis #Elliptic curve #Elliptic function #Elliptic integral #Elliptic rational functions #Fractional Differential Equations Solutions #Function (biology) #Gamma function #Jacobi elliptic functions #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Pure mathematics #Quadratic growth #Quarter period #Rational function

paper · doi:10.1093/imanum/12.4.519

openalex publication_date 1992/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22

Abstract

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 ≤(n/4) and ≤(n/6), where 1≤m≤3 and 1≤n≤5.

Cited by

• Computational strategies for the Riemann zeta function
• Gamma function [wikipedia]