Classroom Computer Capsules

Abstract

where wiy) is the winding number of the curve y around the point zQ. We will study the case for which fiz) = 1 and z0 = 0; even in this case, students have problems with the idea that the integral of 1/(2iriz) around a closed curve should be an integer that counts the number of trips around the origin. The library of demo programs in fiz) includes one that is intended to illustrate the Cauchy integral formula, but it is not especially effective as the first illustration. We assume that you will prepare your own demonstration page with the features shown in Figure 1: separate screens to show the z plane, the integrand, and the integral, suitably positioned, sized, scaled, and labeled, with five circles selected in the domain, some of which contain the origin, and some of which do not. The tic marks on the axes in each picture are at one unit from 0 in each direction. The preparation requires one additional step: In the Circles menu, change the default n-gon approximation to each circle to n = 720. This will reduce the step size in the numerical integration (Euler's method) enough to make the graphical answer really look like an integer. Lesson. Load your Cauchy integral formula page, and press the draw key. The dynamics of the drawing are at least as important as the finished product. The five circles are drawn one at a time, smallest first. As each circle is drawn in the left picture, the inversion appears simultaneously in the right picture and the integral