Applying Rouché's Theorem

Complex Analysis · Axiom Academy

EXAMPLE Applying Rouché's Theorem Counting the zeros of a polynomial inside an annulus by applying Rouché on two circles How many zeros does p(z) = z^4 - 6z + 3 have in the annulus ? We never solve the quartic. We count the zeros inside the big circle |z|=2 , count the zeros inside the small circle |z|=1 , and subtract. Rouché's theorem gives each count without finding a single root. (The dots show where the four zeros actually fall — one green zero inside the small disk, three orange zeros in the ring — but the whole point is that we get the counts without locating them.) Nice work. You counted the zeros of z^4 - 6z + 3 in a ring without solving anything — by applying Rouché's theorem twice and subtracting. The recipe: write p = f + g where f has zeros you already know, check on the contour, then p and f have the same number of zeros inside. The dominant term changes with the radius: on a large circle the highest-degree term z^n wins; on a small circle the lowest-degree non-constant term (here the linear -6z ) wins. Pick f to be whichever term is biggest there. Annulus = subtraction: zeros in equal (zeros in ) minus (zeros in ) = 4 - 1 = 3 . Common choices of f : the leading term z^n for large circles; a constant for tiny circles (it has no zeros, proving none inside); a single linear or middle term when it dominates at an intermediate radius.

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