Applying the Argument Principle

Complex Analysis · Axiom Academy

EXAMPLE Applying the Argument Principle Count the zeros of a polynomial inside a circle — without finding a single root Show that f(z) = z^4 - 6z + 3 has exactly one zero inside the unit circle |z| = 1 . We will count the zeros without solving the quartic — using the argument principle and its workhorse corollary, Rouché's theorem. Nice work — you counted the roots of a quartic inside a disk without finding any of them. The moves worth keeping: The argument principle counts by winding: the number of enclosed zeros equals how many times circles the origin, . Rouché is the shortcut: if one term g strictly dominates the rest on the contour ( ), then f has the same number of zeros as g — no integral needed. Find the dominant term on the contour: on |z|=1 we had |z^4|=1 , | -6z |=6 , |3|=3 , so -6z wins ( ). Count the easy zeros: g(z)=-6z has a single zero ( z=0 ) inside |z|=1 , so f does too. Result: z^4-6z+3 has exactly 1 zero inside |z|=1 . (Its real root is — but we never had to find it.) This is the engine behind a clean proof of the Fundamental Theorem of Algebra and behind locating roots for stability analysis — counting zeros region by region, purely from the size of the terms.

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