Counting Zeros and Poles

Complex Analysis · Axiom Academy

Without ever solving for one — a single contour integral counts every zero and pole a function has inside a loop. Counting things you never have to find Here is a question that sounds impossible: given a function f(z) and a closed loop , how many of its zeros and poles sit inside the loop — without solving a single equation? Complex analysis answers it with one contour integral. Watch the trick happen first, then drive it yourself. As the point z travels once around the loop on the left, its image w = f(z) traces a path on the right. Keep your eye on how many times that image circles the origin — that count is the answer. Here f(z) = z^3 has a triple zero at the origin and no poles, so N - P = 3 — and the image circles the origin exactly three times. More zeros inside, more times around Try f(z) = z^ k , which stacks k zeros at the origin. Drag the slider to change k and watch the image loop: it circles the origin exactly k times. The winding count keeps perfect track of how many zeros are inside. Zeros count positively : each one adds a full turn to the image, so the winding number reads off N directly. Now add poles. With Z zeros and P poles inside the loop, the image winds a net Z - P times. Poles wind the image backward — and if zeros and poles balance, the image never encircles the origin at all. Set both counts and watch the net winding. Poles count negatively . The single integral returns this signed total N - P in one stroke. The capstone of complex analysis

This is the written version of the interactive lesson above. See the full Complex Analysis course.