Rouché's Theorem

Complex Analysis · Axiom Academy

Count the zeros of a hard function by comparing it to an easy one — small perturbations cannot change the winding count. Before Rouché, meet its engine. As z travels once around a closed contour , the image point f(z) traces its own closed curve in the w -plane. The argument principle says the number of times that image curve winds around the origin is exactly the zeros minus the poles of f inside . zeros - poles inside = how many times circles 0 Here f(z) = z^2 - 1 has zeros at , both inside the circle |z| = 2 , and no poles. So the image must wind around 0 exactly twice — and it does. 2. Rouché: A Small Perturbation Can't Move the Count Now the theorem itself. Suppose f and g are analytic inside and on a simple closed contour , and on the function g stays strictly smaller than f : Then f and f + g have the same number of zeros inside (counted with multiplicity). Slide a dial t from 0 to 1 and watch the homotopy . On we have , so h_t can never reach 0 on the contour. Its image curve stretches and wobbles, but because it never crosses the origin its winding number — the zero count — is frozen. The headline payoff: every polynomial has a root Take , with f(z) = z^n and g the lower-order tail. On a big circle |z| = R , the top term swamps the rest: . By Rouché, p has the same number of zeros inside as z^n — namely n . That is the fundamental theorem of algebra , in one stroke. 3. Read Off the Count: z^4 + 6z + 3 in

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