Complex standard functions and their implementation in the CoStLy library

Abstract

The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function f and a given rectangular complex interval z , some interval w is computed that contains all function values of f in z . This is done by expressing the real and the imaginary part of f as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that w is the smallest rectangular interval containing the range of f . The algorithms presented in this article have been implemented in a C++ class library called CoStLy ( C omplex S tandard Functions L icense), which is distributed under the conditions of the GNU General Public License.