ABSTRACT
Several of the references describe quadrature rules for numerically approximating a contour integral. The numerical integrator in the language Mathematica allows an arbitrary path in the complex place to be specified. For example, consider the integral in (30.5). The integration contour may be deformed to the square with vertices at { ±1, ±i} without changing the value of the integral. If Mathematic& is input "Nintegrate [-2I/(z-2+4z+1) ,z,1, I, -1,-I ,1]", then the result is 3. 6276 + 0. I, which is the correct numerical value. (Here I is used to represent i.) The theory of multidimensional residues is based on the general Stokes formula and its corollary, the Cauchy-Poincare integral theorem. Serious topological difficulties arise for analytic functions of several complex variables because the role of the singular point is now played by surfaces in en. These surfaces generally have a complicated structure requiring tools such as algebraic topology.
