Computing Optimal Boxes Enclosing the Range of Complex Functions Walter Kraemer Bergische Universitaet Wuppertal Department of Mathematics and Computer Science Scientific Computing/Software Engineering Gaussstrasse 20 42119 Wuppertal, Germany kraemer@math.uni-wuppertal.de There are two reasonable choices to define the sets of complex numbers that will constitute easy to handle complex intervals: a) rectangles as complex intervals with sides parallel to the coordinate axes (also called boxes) as well as b) circular regions (also called circular disks). In order to be able to compute reliable enclosures for ranges of values of complex expressions using complex intervals it is not only necessary to realize the basic arithmetic operations +, -, *, / for such sets but also to realize elementary functions like sin, cos, log, arsinh and so on. The goal is to get best possible enclosures (boxes or disks depending on the kind of basic interval sets chosen) containing the set of all point results of an arithmetic operation or the set of all function values when evaluated over all points of a complex argument interval. The main topic of the talk is the computation of complex interval functions over rectangles. Computer algebra packages allow to visualize the true ranges of elementary operations and complex functions. Some examples are given. To get optimal complex interval enclosures several approaches are used. If the function is a so called separable function (like the exponential function) real interval arithmetic can be used to get best possible real interval enclosures for the real and imaginary part. The Cartesian product of these intervals describes the optimal box containing all complex function values. Another approach uses partial derivatives along the sides of the complex argument interval to find maximum and minimum of the real and imaginary part functions. Here it is used that the real and imaginary part functions are harmonic functions. This fact allows the application of the Maximum principle reducing the search region for extremal points to the boundary of the complex argument interval. To succeed with this approach it is necessary that the real and imaginary part functions are expressed as functions from R^2 to R. Here again computer algebra packages may be utilized to find appropriate functional expressions for the real and imaginary parts (often several different formulas have to be used to overcome cancellation problems in different subregions). Hints are given how reliable enclosures may be computed taking into account rounding errors as well as errors caused by coordinates of extremal points which are not representable in the complex floating point screen. To show the power of the approach several numerical examples are presented based on a quite new implementation of a comprehensive set of arbitrary precision complex interval functions. Also the results of some time measurements are shown.