Proof of the residue formula at a simple pole

Builds on Holomorphic function Order of a pole Residue Existence of Laurent expansions on an annulus Holomorphic functions are analytic Power series are holomorphic in the disc of convergence Disc of convergence Laurent series

Proof

A lemma. Let $\varphi$ be holomorphic on a disc $D_\delta = \set{z : \abs{z - c} < \delta}$ with $\delta \le r$ , and let $f(z) = \varphi(z)/(z - c)$ for $0 < \abs{z - c} < \delta$ . We show $\operatorname{Res}(f, c) = \varphi(c) = \lim_{z \to c} (z - c) f(z)$ .

By holomorphic functions are analytic, $\varphi(z) = \sum_{n \ge 0} b_n (z - c)^n$ on $D_\delta$ , with $b_0 = \varphi(c)$ .
Dividing by $z - c$ gives $f(z) = \sum_{n \ge -1} b_{n+1} (z - c)^n$ for $0 < \abs{z - c} < \delta$ . This is a Laurent series with coefficient $b_0$ at $(z - c)^{-1}$ .
The Laurent expansion of $f$ on $0 < \abs{z - c} < r$ also converges to $f$ on the smaller annulus $0 < \abs{z - c} < \delta$ .
By the uniqueness part of the existence of Laurent expansions on $0 < \abs{z - c} < \delta$ , the two expansions have the same coefficients.
By the definition of the residue, $\operatorname{Res}(f, c) = b_0 = \varphi(c)$ .
Also $(z - c) f(z) = \varphi(z) \to \varphi(c)$ as $z \to c$ , since $\varphi$ is continuous. This proves the lemma.

The first formula.

Since $c$ is a pole of order $1$ , there are $\delta > 0$ and $\varphi$ holomorphic on $D_\delta$ with $f(z) = \varphi(z)/(z - c)$ for $0 < \abs{z - c} < \delta$ . We may take $\delta \le r$ .
By the lemma, $\operatorname{Res}(f, c) = \lim_{z \to c} (z - c) f(z)$ .

The formula for $g/h$ .

Let $g$ and $h$ be holomorphic on a disc around $c$ , with $g(c) \ne 0$ , $h(c) = 0$ and $h'(c) \ne 0$ , and let $f = g/h$ .
By analyticity, $h(z) = \sum_{n \ge 1} d_n (z - c)^n$ on a disc around $c$ , with $d_1 = h'(c)$ by term-by-term differentiation.
The series $k(z) = \sum_{n \ge 1} d_n (z - c)^{n-1}$ converges on the same disc, since for $z \ne c$ its partial sums are those of $h(z)$ divided by $z - c$ . Its radius of convergence is at least the radius of the disc, because a convergent series has bounded terms.
By power series are holomorphic, $k$ is holomorphic there. We have $h(z) = (z - c) k(z)$ and $k(c) = h'(c) \ne 0$ .
By continuity, $k \ne 0$ and $g$ is defined on some disc $D_\delta$ with $\delta \le r$ . On it, $\varphi = g/k$ is holomorphic, and $f(z) = \varphi(z)/(z - c)$ for $0 < \abs{z - c} < \delta$ .
By the lemma, $\operatorname{Res}(f, c) = \varphi(c) = g(c)/h'(c)$ .

Idea

Near a simple pole, $f$ is a holomorphic function $\varphi$ divided by $z - c$ . Dividing the Taylor series of $\varphi$ by $z - c$ gives the Laurent series of $f$ , whose coefficient at $(z - c)^{-1}$ is $\varphi(c)$ . For $f = g/h$ , factor $h(z) = (z - c) k(z)$ with $k(c) = h'(c)$ , and take $\varphi = g/k$ .

Reading path

To understand this page from the ground up, read these 529 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, 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 that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, 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 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 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 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 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 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.

Basis concept
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Proposition concept
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p-metric on R^n example
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Proof that i squared is minus one proof
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Total derivative concept
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p-norm on R^n example
Pascal's rule statement
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Empty set object
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i squared is minus one statement
Partial derivative concept
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Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
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Completeness axiom statement
Integral of 1/z around the unit circle example
Proof of the factorial formula for binomial coefficients by induction proof
Proof of the equality of complex numbers proof
Cartesian product definition
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Differentiable function of several variables definition
Field definition
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Archimedean property statement
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Proof of the commutativity of addition by induction proof
Mathematical proof concept
Order on the natural numbers definition
Propositional logic concept
Proof of the derivative as a little-o approximation proof
The rationals form an ordered field statement
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Proof of the sum of a finite geometric series proof
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The sequence r^n tends to infinity for r > 1 example
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Proof of the derivative of a map into R^n by components proof
Topological space definition
Proof that a composition of continuous functions is continuous proof
Order on the reals definition
Derivative as a little-o approximation statement
Derivative function definition
Definite integral definition
Commutativity of addition statement
Tautology definition
Proof of strong induction from ordinary induction proof
Upper and lower Darboux integrals definition
Uniform convergence definition
Sum of a finite geometric series statement
Radius of convergence definition
Convergent series definition
Sine and cosine definition
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Proof of the binomial theorem by induction proof
Sequential compactness definition
Interval definition
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Proof that the complex numbers form a field proof
Subspace topology definition
Function definition
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Logical equivalence definition
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Continuously differentiable function definition
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Proof of the epsilon-delta form of continuity proof
Image of a set definition
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The complex numbers form a field statement
Metric space definition
Proof of the well-ordering principle from strong induction proof
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Partial order definition
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Axiom of choice statement
Epsilon-delta definition of a limit statement
Left inverse definition
Right inverse definition
Surjective function definition
Epsilon-delta definition of continuity statement
Closed ball definition
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Lipschitz map definition
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Triangle inequality for a metric statement
Double negation law statement
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Well-ordering principle statement
Maximal element definition
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Upper bound definition
Uniform continuity definition
Composition of continuous functions is continuous statement
Bijective function definition
Metric topology definition
Bounded set definition
Taylor series definition
Proof by contradiction concept
Proof of the diameter characterisation of Cauchy sequences proof
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Uniform continuity on a set definition
Open set definition
Proof that a function is bijective iff it has a two-sided inverse proof
Maclaurin series definition
Epsilon characterisation of Cauchy sequences statement
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
Bijective iff it has a two-sided inverse statement
Closed set definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Open cover definition
Continuous map of topological spaces definition
Zorn's lemma statement
Equivalent metrics definition
Proof of the description of the open sets of a metric space proof
Complete metric space definition
Lipschitz functions are uniformly continuous statement
The reals form an ordered field statement
Proof that a set is open if and only if its complement is closed proof
Compact space definition
Open sets characterised by neighbourhoods statement
Equivalent norms definition
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
Continuous linear map definition
Compact set definition
Banach space definition
Continuity at a point of a topological space definition
Proof that closed sets contain their limits proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Isomorphism with the coordinate space statement
Proof of the epsilon-N definition of the limit of a sequence proof
Every spanning set contains a basis statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Closed sets contain their limits statement
Proof of continuity via closed sets proof
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open 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
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
Proof of the reverse triangle inequality 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
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of continuity via open sets in metric spaces proof
Proof that metric spaces are Hausdorff 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
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
The sequence 1/n tends to zero example
Proof of the term test proof
Proof that Cauchy sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Proof that compact subsets of a Hausdorff space are closed proof
Proof of Cantor's intersection theorem proof
Annulus definition
Reverse triangle inequality statement
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Proof of the sequence characterisation of uniform continuity proof
Differentiable functions are continuous statement
Linearity of the integral statement
Proof of the triangle inequality for norms proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the ratio formula for the radius of convergence proof
Proof of the monotonicity of the integral proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
Compact subsets of a Hausdorff space are closed statement
Cantor's intersection theorem statement
A complex number times its conjugate statement
Continuous image of a compact set is compact statement
Proof of the Cauchy–Riemann equations proof
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
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
Term test statement
Triangle inequality for norms statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Isolated singularity definition
Zero of a holomorphic function definition
Cauchy–Riemann equations statement
Proof that the modulus is multiplicative proof
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof of the Heine–Cantor theorem by a finite cover of balls proof
Proof that absolutely convergent series in a Banach space converge 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
Geometric series example
Proof of Riemann's criterion for integrability proof
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Pole definition
Modulus is multiplicative statement
Heine–Cantor theorem statement
Absolutely convergent series in a Banach space converge statement
Lipschitz maps are uniformly continuous statement
Proof of Euler's formula by power series proof
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Proof of the root test proof
Proof of the Cauchy–Hadamard theorem proof
Proof of the ratio test proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Euler's formula statement
Bounded linear maps are continuous statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Comparison test statement
Cauchy product of absolutely convergent series statement
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Proof of the derivative of a map into a product proof
Chain rule statement
Product of continuous functions is continuous statement
Additivity of the integral over intervals statement
Proof of the product of exponentials by the Cauchy product proof
Proof of the Heine–Borel theorem by sequential compactness proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Product rule statement
Ratio test statement
Linearity of differentiation statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Product of exponentials statement
Derivative of a map into a product statement
Heine–Borel theorem statement
Proof that power series are holomorphic in the disc of convergence proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the constant multiple rule proof
Boundedness theorem statement
Proof of the power rule by induction proof
Proof of the first fundamental theorem of calculus proof
Ratio formula for the radius of convergence statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Power series are holomorphic in the disc of convergence statement
Derivative of a map into R^n by components statement
Constant multiple rule statement
Power 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
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof that continuous functions are Riemann integrable proof
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Absolutely convergent series converge statement
Equivalence of norms in finite dimension statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the estimation lemma proof
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the mean value theorem for integrals proof
Proof of the Weierstrass M-test proof
Estimation lemma statement
Linear maps on finite-dimensional spaces are continuous statement
Mean value theorem for integrals statement
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
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 a function with zero derivative on an open interval is constant proof
Proof that continuous partial derivatives imply differentiability 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 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
Order of a pole definition
The complex exponential is 2πi-periodic statement
Proof of the existence and uniqueness of Laurent expansions on an annulus proof
Existence of Laurent expansions on an annulus statement
Residue definition
Proof of the residue formula at a simple pole proof

Where this leads

14 pages build on this one.

Tag	`BSAK6Z40XW`
URI	`m2://BSAK6Z40XW`
Kind	proof
Writing	W0 stub
MSC	30E20
Proves	Residue at a simple pole
Uses	Holomorphic function, Order of a pole, Residue, Existence of Laurent expansions on an annulus, Holomorphic functions are analytic, Power series are holomorphic in the disc of convergence, Disc of convergence, Laurent series
Changed	4 November 2026

Proof of the residue formula at a simple pole

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