A MATLAB-based visualization package for complex functions, and their mappings and integrals

This paper aims to provide a tutorial on the visualization of complex functions. The MATLAB-based package ComplexGUI was prepared to illustrate mappings by elementary complex functions. The tool shows how user-defi ned curves and regions are transformed by some elementary functions. Concepts such as branch points, branch cuts, the principal branch of multivalued functions, and Riemann surfaces are emphasized. The package also enables asymptotic evaluation of Bessel integrals along user-specifi ed contours, and compares the result with the built-in function in MATLAB. The tool allows deforming the integration contour to a path through the saddle points in the direction of the steepest-decent path to obtain a better result.