Proof that holomorphic functions are analytic

Builds on Holomorphic function Power series in the complex plane Cauchy's integral formula Circle integral Sum of a finite geometric series Heine–Borel theorem Continuous real function on a compact set attains its bounds Estimation lemma The sequence r^n tends to zero for |r| < 1 Triangle inequality for norms

Proof

We expand the Cauchy kernel in a geometric series and integrate term by term.

Fix $\rho$ with $0 < \rho < r$ . The closed disc $\set{w : \abs{w - c} \le \rho}$ lies in $D \subseteq U$ . For $n \ge 0$ put $a_n(\rho) = \frac{1}{2 \pi i} \oint_{\abs{w - c} = \rho} \frac{f(w)}{(w - c)^{n+1}} \, dw$ , a circle integral of a continuous function.
Fix $z$ with $\abs{z - c} < \rho$ and put $q = \abs{z - c}/\rho < 1$ . By Cauchy's integral formula, $f(z) = \frac{1}{2 \pi i} \oint_{\abs{w - c} = \rho} \frac{f(w)}{w - z} \, dw$ .
For $w$ on the circle put $u = (z - c)/(w - c)$ , so $\abs{u} = q$ and $w - z = (w - c)(1 - u)$ . By the finite geometric sum, $\frac{1}{1 - u} = \sum_{n=0}^{N} u^n + \frac{u^{N+1}}{1 - u}$ .
Hence $\frac{f(w)}{w - z} = \sum_{n=0}^{N} \frac{f(w)}{(w - c)^{n+1}} (z - c)^n + T_N(w)$ , with $T_N(w) = \frac{f(w) \, u^{N+1}}{(w - c)(1 - u)}$ .
The circle is closed and bounded, hence compact, so $\abs{f} \le M$ on it for some $M$ by the extreme value theorem. Since $\abs{1 - u} \ge 1 - q$ by the triangle inequality, $\abs{T_N(w)} \le \frac{M q^{N+1}}{\rho (1 - q)}$ .
Integrate step 4 over the circle and divide by $2 \pi i$ . We get $f(z) = \sum_{n=0}^{N} a_n(\rho) (z - c)^n + \frac{1}{2\pi i} \oint T_N(w) \, dw$ .
The circle has length $2 \pi \rho$ . By the estimation lemma, the last term has modulus at most $\frac{M q^{N+1}}{1 - q}$ . This tends to $0$ as $N \to \infty$ , since $q^N \to 0$ .
So the power series $\sum_{n \ge 0} a_n(\rho) (z - c)^n$ converges to $f(z)$ whenever $\abs{z - c} < \rho$ .
Now let $0 < \rho_1 < \rho_2 < r$ and $b_n = a_n(\rho_1) - a_n(\rho_2)$ . By step 8 for both radii, $\sum_{n \ge 0} b_n w^n = 0$ for every $w$ with $\abs{w} < \rho_1$ .
Suppose some $b_n \ne 0$ , and let $m$ be the least such $n$ . Put $s = \rho_1 / 2$ . The series $\sum b_n s^n$ converges, so its terms tend to $0$ . Hence $\abs{b_n} s^n \le K$ for all $n$ , for some $K$ .
Let $0 < \abs{w} < s$ . Then $\abs{b_n w^{n-m}} \le K s^{-m} (\abs{w}/s)^{n-m}$ for $n \ge m$ . So $T(w) = \sum_{n > m} b_n w^{n-m}$ converges and $\abs{T(w)} \le K s^{-m} \frac{\abs{w}/s}{1 - \abs{w}/s}$ .
By steps 9 and 10, $0 = \sum_{n \ge m} b_n w^n = w^m (b_m + T(w))$ . Since $w \ne 0$ , $b_m = -T(w)$ .
Letting $w \to 0$ in step 11 gives $\abs{b_m} \le \lim \abs{T(w)} = 0$ . This contradicts $b_m \ne 0$ . So $a_n(\rho_1) = a_n(\rho_2)$ for all $n$ .
Thus $a_n = a_n(\rho)$ does not depend on $\rho \in (0, r)$ , and it is given by the stated formula.
Let $z \in D$ . Choose $\rho$ with $\abs{z - c} < \rho < r$ . By steps 8 and 14, $\sum_{n \ge 0} a_n (z - c)^n$ converges to $f(z)$ .
Finally, let $c \in U$ be any point. Since $U$ is open, some disc $\set{z : \abs{z - c} < r}$ with $r > 0$ lies in $U$ . By step 15, $f$ is the sum of a convergent power series near $c$ . So $f$ is analytic on $U$ .

Idea

Cauchy's formula writes $f(z)$ as an average of $f(w)/(w - z)$ over a circle. The kernel $1/(w - z)$ is a geometric series in $(z - c)/(w - c)$ , which converges uniformly on the circle. Integrating term by term gives the power series, and the uniqueness of coefficients shows they do not depend on the radius.

Reading path

To understand this page from the ground up, read these 324 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, 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 that open sets are the sets that are neighbourhoods of all their points, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, 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 with an auxiliary difference quotient, Proof of the reverse triangle inequality, Proof of the derivative as a little-o approximation, 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 the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, 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 that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof that the absolute value is multiplicative, 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 the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, 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 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 of the existence of the infimum by negation, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of integration by substitution, Proof that metric spaces are Hausdorff, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, 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 that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that open balls are open, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that continuous partial derivatives imply differentiability, Proof of the fundamental theorem of calculus, Proof of the mean value theorem from Rolle's theorem, 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.

Complex numbers object
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Integers object
Limit point of a set definition
Lipschitz continuity definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Norm definition
Circle integral definition
Empty set object
Family of sets definition
Ordered pair definition
Limit of a function of several variables concept
Partial derivative concept
Intersection definition
Union definition
Addition of natural numbers definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
Riemann integrable function definition
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Completeness axiom statement
Normed vector space definition
Integral of 1/z around the unit circle example
Cartesian product definition
Differentiable function of several variables definition
Partial derivative definition
Jacobian matrix concept
Continuous function definition
Union of a family of sets definition
Antiderivative definition
Differentiable function definition
One-sided derivative definition
Order on the natural numbers definition
Proof of the commutativity of addition by induction proof
Archimedean property statement
Subset definition
Riemann integral definition
The rationals form an ordered field statement
Proof of the derivative as a little-o approximation proof
Propositional logic concept
Mathematical proof concept
Upper and lower Darboux sums definition
Uniqueness quantifier definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Sequential continuity definition
Proof of the derivative of a map into R^n by components proof
Topological space definition
Binary relation definition
Proof that a composition of continuous functions is continuous proof
Order on the reals definition
Universal set definition
Derivative function definition
Commutativity of addition statement
Derivative as a little-o approximation statement
Proof of strong induction from ordinary induction proof
Definite integral definition
Tautology definition
Upper and lower Darboux integrals definition
Sum of a finite geometric series statement
Sequential compactness definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Higher derivative definition
Subspace topology definition
Complement definition
Strong induction statement
Function definition
Logical equivalence definition
Preorder definition
Continuously differentiable function definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Proof of the well-ordering principle from strong induction proof
Metric space definition
Image of a set definition
Preimage of a set definition
Proof of the double negation law by a truth table proof
Partial order definition
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Well-ordering principle statement
Diameter of a set definition
Double negation law statement
Total order definition
Bounded set definition
Metric topology definition
Composition of continuous functions is continuous statement
Uniform continuity definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Open set definition
Uniform continuity on a set definition
Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Hausdorff space definition
Equivalent metrics definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Closed set definition
Continuous map of topological spaces definition
Proof of the description of the open sets of a metric space proof
Open cover definition
Lipschitz functions are uniformly continuous statement
The reals form an ordered field statement
Equivalent norms definition
Open sets characterised by neighbourhoods statement
Open sets of a metric space statement
Compact space definition
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the existence of the infimum by negation proof
Continuous linear map definition
Continuity at a point of a topological space definition
Proof that closed sets contain their limits proof
Proof that open balls are open proof
Compact set definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Closed sets contain their limits statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Open balls are open statement
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Modulus of a complex number definition
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof that metric spaces are Hausdorff proof
Proof of continuity via open sets in metric spaces proof
Proof of the reverse triangle inequality proof
Compactness via open covers statement
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of the limit of a sum of sequences proof
Proof of the uniqueness of limits proof
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Intermediate value theorem statement
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a Hausdorff space are closed proof
Metric spaces are Hausdorff statement
Proof of Cantor's intersection theorem proof
Reverse triangle inequality statement
Proof that compact subsets of a metric space are bounded proof
Proof that a continuous image of a compact set is compact proof
Proof that compactness and sequential compactness agree in metric spaces proof
Lebesgue number lemma statement
Proof of the triangle inequality for norms proof
Proof of the sequence characterisation of uniform continuity proof
Differentiable functions are continuous statement
Limit of a sum of sequences statement
Linearity of the integral statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
Proof of the Cauchy–Riemann equations proof
Compact subsets of a Hausdorff space are closed statement
Continuity via open sets statement
Cantor's intersection theorem statement
Compact subsets of a metric space are bounded statement
Compactness and sequential compactness agree in metric spaces statement
Continuous image of a compact set is compact statement
Epsilon-delta uniform continuity statement
Proof of the limit of a product of sequences proof
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Cauchy–Riemann equations statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that Lipschitz maps are uniformly continuous proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Proof of Riemann's criterion for integrability proof
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bounded linear maps are continuous statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Product of continuous functions is continuous statement
Proof of the derivative of a map into a product proof
Chain rule statement
Proof of the Heine–Borel theorem by sequential compactness proof
Additivity of the integral over intervals statement
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Heine–Borel theorem statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the constant multiple rule proof
Boundedness theorem statement
Proof of the first fundamental theorem of calculus proof
Derivative of a map into R^n by components statement
Constant multiple rule statement
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the estimation lemma proof
Proof of the mean value theorem for integrals proof
Estimation lemma statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that continuous partial derivatives imply differentiability proof
Proof that the integral of a continuous function is an antiderivative proof
Proof of the second fundamental theorem of calculus proof
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Proof of integration by substitution proof
Integration by substitution statement
Fundamental theorem of calculus statement
Proof of the fundamental theorem of contour integration proof
Fundamental theorem of contour integration statement
Proof of Goursat's theorem by repeated subdivision proof
Goursat's theorem statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Cauchy's integral formula statement
Proof that holomorphic functions are analytic proof

Where this leads

211 pages build on this one.

Tag	`T37KYMZZS5`
URI	`m2://T37KYMZZS5`
Kind	proof
Writing	W0 stub
MSC	30B10
Proves	Holomorphic functions are analytic
Uses	Holomorphic function, Power series in the complex plane, Cauchy's integral formula, Circle integral, Sum of a finite geometric series, Heine–Borel theorem, Continuous real function on a compact set attains its bounds, Estimation lemma, The sequence r^n tends to zero for \|r\| < 1, Triangle inequality for norms
Changed	4 November 2026

Proof that holomorphic functions are analytic

Proof

Idea

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