Landen transformations applied to approximation
: Kargar, Rahim; Rainio, Oona; Vuorinen, Matti
Publisher: Yokohama Publications
: 2024
: Pure and Applied Functional Analysis
: Pure and Applied Functional Analysis
: 9
: 2
: 503
: 516
: 2189-3756
: https://arxiv.org/pdf/2212.09336(external)
: https://arxiv.org/abs/2212.09336(external)
We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as inverses of such quotients. In particular, we consider the distortion function φk(r) which gives a majorant for |f(x)| when f: B2 → B2, f(0) = 0, is a quasiconformal mapping of the unit disk B2. It turns out that the approximation method is very simple: Five steps of Landen iteration are enough to achieve machine precision.