Proof of Rouché's theorem by the argument principle and continuity

Builds on Holomorphic function Meromorphic function Reverse triangle inequality Heine–Borel theorem Continuous real function on a compact set attains its bounds Holomorphic functions are infinitely differentiable Order of a zero Argument principle Estimation lemma Circle integral

Proof

Let $f$ and $g$ be holomorphic on an open set $W$ that contains $\set{z : \abs{z - c} \le r}$ . Let $C$ be the circle $\abs{z - c} = r$ , and assume $\abs{g} < \abs{f}$ on $C$ .
For $t \in [0, 1]$ , let $F_t = f + t g$ . It is holomorphic on $W$ , so it is meromorphic on $W$ with no poles.
On $C$ , the reverse triangle inequality gives $\abs{F_t} \ge \abs{f} - t \abs{g} \ge \abs{f} - \abs{g}$ .
The circle $C$ is compact by the Heine–Borel theorem. By attainment of bounds, $\abs{f} - \abs{g}$ has a minimum $m$ on $C$ , and $m > 0$ by hypothesis. So $\abs{F_t} \ge m$ on $C$ .
Let $Z(t)$ be the number of zeros of $F_t$ in $\abs{z - c} < r$ , counted with order. By step 4 and the argument principle, $Z(t) = \frac{1}{2\pi i} \oint_C \frac{F_t'}{F_t} \, dz$ , a circle integral. In particular $Z(t)$ is an integer.
For $s, t \in [0, 1]$ , a direct expansion gives $(f' + t g')(f + s g) - (f' + s g')(f + t g) = (t - s)(g' f - f' g)$ .
Dividing by $F_t F_s$ gives $\frac{F_t'}{F_t} - \frac{F_s'}{F_s} = \frac{(t - s)(g' f - f' g)}{F_t F_s}$ on $C$ .
By holomorphic functions are infinitely differentiable, $f'$ and $g'$ are holomorphic, hence continuous. So $\abs{g' f - f' g}$ has a maximum $K_0$ on $C$ .
By steps 4 and 7, $\abs{F_t'/F_t - F_s'/F_s} \le \abs{t - s} K_0 / m^2$ on $C$ . The length of $C$ is $2\pi r$ .
By the estimation lemma and step 5, $\abs{Z(t) - Z(s)} \le \frac{1}{2\pi} \cdot 2\pi r \cdot \frac{\abs{t - s} K_0}{m^2} = \frac{r K_0}{m^2} \abs{t - s}$ .
Choose an integer $q \ge 1$ with $q > r K_0 / m^2$ , and let $t_j = j/q$ for $j = 0, \dots, q$ . By step 10, $\abs{Z(t_{j+1}) - Z(t_j)} < 1$ .
Since the values of $Z$ are integers, $Z(t_{j+1}) = Z(t_j)$ for each $j$ . Hence $Z(0) = Z(1)$ .
Here $F_0 = f$ and $F_1 = f + g$ . So $f$ and $f + g$ have the same number of zeros in $\abs{z - c} < r$ , counted with order.

Idea

Move continuously from $f$ to $f + g$ through $f + t g$ . The condition $\abs{g} < \abs{f}$ on the circle keeps every $f + t g$ away from zero there. So the argument principle counts the zeros of each $f + t g$ by an integral that depends continuously on $t$ . An integer that varies continuously cannot change.

Reading path

To understand this page from the ground up, read these 552 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the argument principle by the residue theorem, Proof of the Cauchy–Hadamard theorem, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the power series of sine, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the reverse triangle inequality, Proof of the derivative as a little-o approximation, Proof that cos pi = -1 and sin pi = 0, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof of differentiation under the integral sign, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof of the residue formula at a simple pole, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof of the residue theorem by subtracting principal parts, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the complex exponential is 2πi-periodic, Proof that the absolute value is multiplicative, Proof of the power series of cosine, Proof of the squeeze theorem for sequences, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof that the continuous image of a connected set is connected, Proof that i squared is minus one, Proof of the linearity of differentiation, Proof of the angle addition formulas by differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof of the Heine–Cantor theorem by a finite cover of balls, Proof that holomorphic functions on a disc have primitives, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the modulus is multiplicative, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that path-connected spaces are connected, Proof that every spanning set contains a basis, by Zorn's lemma, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that connected subsets of the real line are intervals, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof of integration by substitution, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof of the product of exponentials by the Cauchy product, Proof of the second fundamental theorem of calculus, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof that sine and cosine are 2π-periodic with smallest period 2π, Proof of the epsilon-delta form of the limit of a function, Proof of the isolated zeros theorem by power series, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of the Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof of the identity theorem by connectedness, Proof that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the existence and uniqueness of Laurent expansions on an annulus, Proof of the Pythagorean trigonometric identity by differentiation, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that absolutely convergent series in a Banach space converge, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof that a function with zero derivative on an open interval is constant, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof of the ratio formula for the radius of convergence, Proof of Euler's formula by power series, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

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Proof of the product of exponentials by the Cauchy product proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
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Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Derivative of a map into a product statement
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Proof that power series are holomorphic in the disc of convergence proof
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Proof of the first fundamental theorem of calculus proof
Ratio formula for the radius of convergence statement
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Weierstrass M-test statement
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Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof of differentiation under the integral sign proof
Proof that continuous partial derivatives imply differentiability proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that the integral of a continuous function is an antiderivative proof
Chain rule in several variables statement
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof that a function with positive derivative is strictly increasing proof
Proof of the second fundamental theorem of calculus proof
A function with zero derivative is constant statement
Differentiation under the integral sign statement
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Uniform convergence of derivatives allows term-by-term differentiation statement
Second fundamental theorem of calculus statement
A function with positive derivative is strictly increasing statement
Proof of the fundamental theorem of calculus proof
Proof of integration by substitution proof
Proof of term-by-term differentiation proof
Integration by substitution statement
Fundamental theorem of calculus statement
Term-by-term differentiation statement
Proof of the fundamental theorem of contour integration proof
Proof that power series are differentiable inside the radius of convergence proof
Fundamental theorem of contour integration statement
Power series are differentiable inside the radius of convergence statement
Proof of Goursat's theorem by repeated subdivision proof
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
Goursat's theorem statement
Derivative of cosine statement
Derivative of sine statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Proof that holomorphic functions on a disc have primitives proof
Proof of the angle addition formulas by differentiation proof
Proof of the Pythagorean trigonometric identity by differentiation proof
Proof of the power series of cosine proof
Proof of the power series of sine proof
pi object
Cauchy's integral formula statement
Holomorphic functions on a disc have primitives statement
Pythagorean trigonometric identity statement
Power series of cosine statement
Power series of sine statement
Angle addition formulas statement
Proof that holomorphic functions are analytic proof
Proof that cos pi = -1 and sin pi = 0 proof
Holomorphic functions are analytic statement
cos pi = -1 statement
Proof of the isolated zeros theorem by power series proof
Proof that holomorphic functions are infinitely differentiable proof
Proof that sine and cosine are 2π-periodic with smallest period 2π proof
Holomorphic functions are infinitely differentiable statement
Sine is 2π-periodic statement
Order of a zero definition
Proof that the complex exponential is 2πi-periodic proof
Isolated zeros theorem statement
Order of a pole definition
The complex exponential is 2πi-periodic statement
Proof of the identity theorem by connectedness proof
Proof of the existence and uniqueness of Laurent expansions on an annulus proof
Identity theorem statement
Existence of Laurent expansions on an annulus statement
Residue definition
Proof of the residue formula at a simple pole proof
Proof of the residue theorem by subtracting principal parts proof
Residue at a simple pole statement
Residue theorem statement
Proof of the argument principle by the residue theorem proof
Argument principle statement
Proof of Rouché's theorem by the argument principle and continuity proof

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5 pages build on this one.

Tag	`J3PN78KXPF`
URI	`m2://J3PN78KXPF`
Kind	proof
Writing	W0 stub
MSC	30C15
Proves	Rouché's theorem
Uses	Holomorphic function, Meromorphic function, Reverse triangle inequality, Heine–Borel theorem, Continuous real function on a compact set attains its bounds, Holomorphic functions are infinitely differentiable, Order of a zero, Argument principle, Estimation lemma, Circle integral
Changed	4 November 2026

Proof of Rouché's theorem by the argument principle and continuity

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