Rouché's theorem

Rouché's theorem states that holomorphic functions f and g have the same number of zeros, counted with multiplicity, inside a contour if |f−g|<|f| everywhere on the contour.

Argument principle: The argument principle counts zeros minus poles of a meromorphic function inside a contour using a contour integral. Rouché's theorem follows by applying this principle to a continuous family of functions.
Fundamental theorem of algebra: The fundamental theorem of algebra states that every nonconstant complex polynomial has a complex root. Rouché's theorem can locate how many of a polynomial's roots lie in a chosen region.
Complex analysis: Complex analysis studies functions of complex variables and their analytic properties. Rouché's theorem is a central tool in its study of zeros and contours.
Schwarz lemma: The Schwarz lemma bounds holomorphic self-maps of the unit disk that fix the origin. It constrains function values rather than comparing two functions to count zeros.
Winding number: The winding number measures how many times a closed curve encircles a point, with orientation. The theorem preserves the boundary curve's winding around zero during deformation.
Polynomial root location: Polynomial root location studies where a polynomial's zeros lie in the complex plane. The theorem provides a boundary test for counting roots without solving for them.
Open set: An open set is a subset of a topological space containing a neighborhood of each of its points. Holomorphic functions and contour interiors are understood within open domains.
Hurwitz's theorem: Hurwitz's theorem states that a locally uniform limit of zero-free holomorphic functions is zero-free or identically zero. It transfers zero information through limits, while Rouché compares functions on a contour.
Homotopy: A homotopy is a continuous deformation of one mathematical object into another. The inequality prevents the boundary values from reaching zero during this deformation.
Eneström–Kakeya theorem: The Eneström–Kakeya theorem bounds the zeros of polynomials with nonnegative coefficients. Its classical proof applies Rouché's theorem on a suitable circle.

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