In complex analysis, Rouché's theorem tells us that if the complex-valued functions f and g are holomorphic inside and on some closed contour C, with |g(z)| < |f(z)| on C, then f and f + g have the same number of zeros inside C, where each zero is counted as many times as its multiplicity. This theorem assumes that the contour C is simple, that is, without self-intersections, and that it is oriented counter-clockwise. Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...

Proof

Denote h = f + g which is holomorphic, being the sum of two holomorphic functions. From the argument principle, we have that The contour C (black), the zeros of f (blue) and the poles of f (red). ...

where N_h is the number of zeroes of h inside C, P_h is the number of poles, and I_h(C,0) is the winding number of h(C) about 0. Since h is analytic inside and on C, it follows that P_h is zero, and A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ...

One has that h′/h = D log h(z), where D denotes the complex derivative. Keeping in mind that h=f+g, we find

The winding number of 1+g/f over C is zero. This because we supposed that |g(z)| < |f(z)|, so g/f is constrained to a circle of radius 1, and adding 1 to g/f shifts it away from zero, and thus 1 + g/f is constrained to a circle of radius 1 about 1, and C under 1 + g/f cannot wind around 0.

The above then equals

which is N_f or the number of zeros of f.