Rouché's theorem
/ roo-SHAY /
Here is a wonderfully practical idea: if you want to count the roots of a complicated function, replace it on the boundary by a simpler companion that dominates it, and the two have the same number of roots inside. Rouche's theorem makes this rigorous. As long as one piece is strictly bigger than the other all along the contour, the small piece cannot create or destroy any roots inside — it can only nudge them around.
The statement: let f and g be holomorphic on and inside a simple closed contour gamma, and suppose |g(z)| is strictly less than |f(z)| at every point z on gamma. Then f and f + g have the same number of zeros inside gamma, counted with multiplicity. The reason, through the argument principle: as z runs around gamma, the point f(z) + g(z) stays within distance |g| < |f| of f(z), so it can never reach 0 (it never gets as far from f as f is from the origin), and it winds around 0 exactly as many times as f does. Equal winding number means equal zero count. Intuitively, you are walking a dog (f + g) on a leash shorter than your distance to a lamppost (the origin): the dog circles the lamppost the same number of times you do.
This is the everyday tool for locating zeros. To count the roots of a polynomial in a disk, split it into a dominant term f and a remainder g, check |g| < |f| on the circle, and read the answer from f, whose roots you already know. It delivers the slickest proof of the fundamental theorem of algebra (on a large circle the top-degree term dominates all the rest), and Hurwitz's theorem follows from it by a limiting argument. The one thing you must verify honestly is the strict inequality on the whole contour — if |g| equals |f| anywhere on gamma, the image of f + g may touch 0 and the conclusion can fail.
How many zeros does z^4 + 5z + 1 have inside |z| = 2? On that circle |z^4| = 16 dominates |5z + 1| at most 11, so take f = z^4, g = 5z + 1; since |g| < |f| on the circle, f + g has the same 4 zeros inside as z^4 does. All four roots lie in the disk of radius 2.
Dominating term f = z^4 controls the count; the smaller g cannot change the number of interior zeros.
The inequality must be strict and hold everywhere on the contour, not just somewhere. If |g| can equal |f| on gamma, the symmetric form (with |f - g| < |f| + |g|) is the safer statement to invoke.