Proof of the fundamental theorem of contour integration

Builds on Contour integral Holomorphic function Complex differentiability Fundamental theorem of calculus Additivity of the integral over intervals

Proof

We show that $t \mapsto F(\gamma(t))$ has derivative $f(\gamma(t)) \gamma'(t)$ and apply the real fundamental theorem of calculus on each piece.

Choose $a = t_0 < t_1 < \dots < t_m = b$ such that $\gamma$ is continuously differentiable on each $[t_{j-1}, t_j]$ , with one-sided derivatives at the end points.
Let $p \in U$ . Define $E(w) = \frac{F(w) - F(p)}{w - p} - F'(p)$ for $w \in U$ , $w \ne p$ , and $E(p) = 0$ . Since $F$ is complex differentiable at $p$ , $E$ is continuous at $p$ .
For every $w \in U$ , $F(w) - F(p) = (F'(p) + E(w)) (w - p)$ .
Fix $j$ , put $\varphi(t) = F(\gamma(t))$ for $t \in [t_{j-1}, t_j]$ , and fix $t$ there. Put $p = \gamma(t)$ . By step 3, for $s \ne 0$ with $t + s \in [t_{j-1}, t_j]$ , $\frac{\varphi(t+s) - \varphi(t)}{s} = (F'(p) + E(\gamma(t+s))) \frac{\gamma(t+s) - \gamma(t)}{s}$ .
As $s \to 0$ , $\gamma(t+s) \to p$ because $\gamma$ is continuous. So $E(\gamma(t+s)) \to 0$ by step 2. Also the last factor tends to $\gamma'(t)$ .
Hence $\varphi'(t) = F'(\gamma(t)) \gamma'(t) = f(\gamma(t)) \gamma'(t)$ , one-sided at the end points. This is continuous in $t$ , since $f$ , $\gamma$ and $\gamma'$ are continuous on the piece.
Apply the fundamental theorem of calculus to $\operatorname{Re} \varphi$ and $\operatorname{Im} \varphi$ on $[t_{j-1}, t_j]$ . It gives $\int_{t_{j-1}}^{t_j} f(\gamma(t)) \gamma'(t) \, dt = F(\gamma(t_j)) - F(\gamma(t_{j-1}))$ .
By the definition of the contour integral and additivity over intervals, $\int_\gamma f(z) \, dz$ is the sum over $j$ of the integrals in step 7.
The sum telescopes: $\int_\gamma f(z) \, dz = F(\gamma(t_m)) - F(\gamma(t_0)) = F(\gamma(b)) - F(\gamma(a))$ .
If $\gamma$ is closed, then $\gamma(b) = \gamma(a)$ , so $\int_\gamma f(z) \, dz = 0$ .

Idea

Along the path, $F(\gamma(t))$ is an ordinary function of the real variable $t$ . The chain rule says its derivative is the integrand $f(\gamma(t)) \gamma'(t)$ , so the real fundamental theorem of calculus applies.

Reading path

To understand this page from the ground up, read these 181 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 approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the epsilon-delta form of continuity, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that continuous functions on a closed bounded interval are uniformly continuous, 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 of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, 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 of Riemann's criterion for integrability, Proof of the sequential characterization of limits, 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 of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, 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 Archimedean property in an Archimedean ordered field, 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 of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the fundamental theorem of calculus, Proof of the mean value theorem from Rolle's theorem, 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
Integers object
Limit point of a set definition
Local maximum definition
Mathematical induction concept
Natural numbers object
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
Contour integral definition
Real and imaginary parts definition
Set definition
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
Ordered pair definition
Continuity at a point definition
Derivative definition
Addition of natural numbers 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
Universal quantifier definition
Completeness axiom statement
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Cartesian product definition
Continuous function definition
Antiderivative definition
Differentiable function definition
Archimedean property statement
Order on the natural numbers definition
Proof of the commutativity of addition by induction proof
Riemann integral definition
The rationals form an ordered field statement
Subset definition
Mathematical proof concept
Propositional logic concept
Upper and lower Darboux sums definition
Convergent sequence definition
Sequential continuity definition
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Definite integral definition
Tautology definition
Upper and lower Darboux integrals definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Strong induction statement
Logical equivalence definition
Preorder 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
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Well-ordering principle statement
Double negation law statement
Total order definition
Uniform continuity definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
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 triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Existence of square roots of nonnegative reals statement
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum 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 the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Intermediate value theorem statement
Proof of the linearity of the integral proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Linearity of the integral statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Monotone convergence theorem for sequences statement
Holomorphic function definition
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
Proof that a decreasing sequence bounded below converges proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Proof of Riemann's criterion for integrability proof
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Riemann's criterion for integrability statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the linearity of differentiation proof
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Linearity of differentiation statement
Additivity of the integral over intervals statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Proof of the first fundamental theorem of calculus proof
Constant multiple rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem 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 mean value theorem for integrals proof
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 of the second fundamental theorem of calculus proof
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Fundamental theorem of calculus statement
Proof of the fundamental theorem of contour integration proof

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232 pages build on this one.

Tag	`R4HZTBV9VJ`
URI	`m2://R4HZTBV9VJ`
Kind	proof
Writing	W0 stub
MSC	30E20
Proves	Fundamental theorem of contour integration
Uses	Contour integral, Holomorphic function, Complex differentiability, Fundamental theorem of calculus, Additivity of the integral over intervals
Changed	4 November 2026

Proof of the fundamental theorem of contour integration

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