# Residue theorem

The residue theorem (Cauchy's residue theorem) is a result of [complex analysis](https://www.edgepedia.org/complex-analysis) stating that the integral of a function around a positively oriented, simple closed contour equals \( 2\pi i \) times the sum of the residues of the function at the isolated singularities enclosed by the contour. It extends [Cauchy's integral theorem](https://www.edgepedia.org/cauchys-integral-theorem), which applies when the integrand is analytic throughout the enclosed region, to the case where the integrand has finitely many isolated singular points, each contributing a term proportional to its residue.<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup><sup> • </sup><sup>[2](https://www.cambridge.org/highereducation/books/introduction-to-complex-variables-and-applications/34E6AB639B24E9193860CBBA32AB72F8/residue-calculus-and-applications-of-contour-integration/BF299B83D54B571567F9478B9A1AB569)</sup>

| Key fact | Detail |
|---|---|
| Residue | If \( f \) has a nonremovable isolated singularity at \( z_0 \), the residue is the coefficient \( a_{-1} \) of \( (z-z_0)^{-1} \) in the Laurent series \( f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n \).<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup> |
| Theorem statement | For a simply connected domain \( D \), a positively oriented simple closed contour \( C \subset D \), and \( f \) analytic on and inside \( C \) except at \( z_1, \ldots, z_n \) inside \( C \): \( \int_C f(z)\,dz = 2\pi i \sum_{k=1}^{n} \mathrm{Res}[f, z_k] \).<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup> |
| Simple pole formula | \( \mathrm{Res}[f, z_0] = \lim_{z \to z_0} (z - z_0) f(z) \); if \( f = g/h \) with \( g \) analytic and \( h \) having a simple zero at \( z_0 \), then \( \mathrm{Res}[f, z_0] = g(z_0)/h'(z_0) \).<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup><sup> • </sup><sup>[3](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_21.html)</sup> |
| Pole of order \( k \) | \( \mathrm{Res}[f, z_0] = \frac{1}{(k-1)!} \lim_{z \to z_0} \frac{d^{k-1}}{dz^{k-1}} \left[ (z - z_0)^k f(z) \right] \).<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup> |
| General contour version | For a closed contour \( \Gamma \) that need not be simple: \( \int_\Gamma f = 2\pi i \sum_k \nu(\Gamma, z_k)\, \mathrm{Res}(f, z_k) \), where \( \nu \) is the winding number.<sup>[4](https://www.math.cuhk.edu.hk/course_builder/1516/math2230a/Notes16.pdf)</sup> |
| Historical origin | Cauchy defined the residue in his 1826 paper in the first number of the *Exercices de Mathématiques*; the residue result for poles arose from his 1825 mémoire and was extended to general isolated singularities by P. A. Laurent in 1843.<sup>[5](https://www.cambridge.org/core/books/abs/cauchy-and-the-creation-of-complex-function-theory/calculus-of-residues/47BA5BADA5CCDA7531A3008F15EE6A70)</sup><sup> • </sup><sup>[6](https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf)</sup> |

## The residue

A function \( f \) with a nonremovable isolated singularity at \( z_0 \) admits a [Laurent series](https://www.edgepedia.org/laurent-series), a two-sided power series in powers of \( z - z_0 \), valid in some punctured disk around \( z_0 \). The coefficient of \( (z - z_0)^{-1} \) in that series is the residue of \( f \) at \( z_0 \), written \( \mathrm{Res}[f, z_0] = a_{-1} \).<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup> The name reflects the fact that this single coefficient is what survives integration: integrating a Laurent series term by term around a circle centered at \( z_0 \), every term integrates to zero except \( a_{-1}/(z - z_0) \), whose integral is \( 2\pi i a_{-1} \).<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup>

If the singularity is removable, all negative-index coefficients vanish and the residue is 0. At a pole of order \( k \), multiplying by \( (z - z_0)^k \), differentiating \( k - 1 \) times, and taking the limit isolates \( a_{-1} \), giving the derivative formula in the table above; the simple pole case is the \( k = 1 \) instance.<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup> For quotients, if \( g \) and \( h \) are analytic at \( z_0 \) and \( h \) has a simple zero there, then \( \mathrm{Res}_{z_0}(g/h) = g(z_0)/h'(z_0) \); for example \( \mathrm{Res}_i \frac{1}{z^2+1} = \frac{1}{2i} \).<sup>[3](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_21.html)</sup>

At an essential singularity, a pole of infinite order, the theorem itself still holds, but the derivative formula does not apply; the practical route is to write the Laurent series directly and read off \( a_{-1} \).<sup>[3](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_21.html)</sup> For example, \( e^{1/z} = 1 + \frac{1}{z} + \frac{1}{2!z^2} + \cdots \) has residue 1 at the origin, so its integral around any positively oriented simple closed contour enclosing 0 is \( 2\pi i \).<sup>[7](https://ximera.osu.edu/math/complexBook/complexBook/complexResidue/complexResidue)</sup>

## Statement and proof idea

Let \( D \) be a simply connected domain, meaning a domain without holes, and let \( C \) be a simple closed positively oriented contour in \( D \). If \( f \) is analytic on and inside \( C \) except at the points \( z_1, \ldots, z_n \) inside \( C \), then

\[ \int_C f(z)\,dz = 2\pi i \sum_{k=1}^{n} \mathrm{Res}[f, z_k]. \]

The proof walls off each singularity with a small positively oriented circle \( C_k \) around \( z_k \), chosen small enough that the circles are mutually disjoint and lie inside \( C \). The extended Cauchy-Goursat theorem gives \( \int_C f\,dz = \sum_k \int_{C_k} f\,dz \), and each small-circle integral equals \( 2\pi i \, \mathrm{Res}[f, z_k] \) because \( z_k \) is the only singularity inside \( C_k \).<sup>[1](https://complexanalysis.org/web/sec_residue-thm.html)</sup> The connection between residues and integration comes from Laurent's theorem, which expresses \( a_{-1} \) itself as \( \frac{1}{2\pi i} \) times a small circular integral of \( f \).<sup>[8](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/22-23/29142/residues.html)</sup>

The simple-contour statement assumes each enclosed singularity is circled once counterclockwise. The general version drops both restrictions: for any closed contour \( \Gamma \) in a domain where \( f \) is analytic except at \( z_1, \ldots, z_m \),

\[ \int_\Gamma f = 2\pi i \sum_{k=1}^{m} \nu(\Gamma, z_k)\, \mathrm{Res}(f, z_k), \]

where the winding number \( \nu(\Gamma, z_k) \) counts how many times, and in which direction, \( \Gamma \) circles \( z_k \).<sup>[4](https://www.math.cuhk.edu.hk/course_builder/1516/math2230a/Notes16.pdf)</sup>

## Evaluating real integrals

The theorem's principal application is the evaluation of definite real integrals, for which it is sometimes the only practical method.<sup>[2](https://www.cambridge.org/highereducation/books/introduction-to-complex-variables-and-applications/34E6AB639B24E9193860CBBA32AB72F8/residue-calculus-and-applications-of-contour-integration/BF299B83D54B571567F9478B9A1AB569)</sup> The general recipe is to extend the real integrand to a complex function, close the integration path with a curve (typically a large semicircle in the upper half-plane), evaluate the whole closed contour by residues, and show that the added curved part vanishes as its radius grows, usually by an ML estimate or Jordan's lemma.<sup>[3](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_21.html)</sup><sup> • </sup><sup>[9](https://unseel.com/mathematics/residue-theorem)</sup>

**Rational integrands.** For \( \int_{-\infty}^{\infty} \frac{dx}{x^2+1} \), close the segment \( [-N, N] \) with the semicircular arc \( z = Ne^{i\theta} \), \( 0 \le \theta \le \pi \). The only enclosed singularity is \( z = i \), with residue \( \frac{1}{2i} \), so the closed-contour integral is \( 2\pi i \cdot \frac{1}{2i} = \pi \). On the arc, \( |1/(z^2+1)| \le 1/N^2 \) while the arc has length \( \pi N \), so its contribution is at most \( \pi/N \) and vanishes as \( N \to \infty \); the real integral is therefore \( \pi \).<sup>[3](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_21.html)</sup> Adding an oscillatory factor works the same way when it is bounded in the upper half-plane: \( e^{iz} = e^{ix - y} \) has modulus \( e^{-y} \le 1 \) there, and the same contour gives \( \int_{-\infty}^{\infty} \frac{\cos x}{x^2+1}\,dx = \frac{\pi}{e} \).<sup>[10](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_22.html)</sup>

**Trigonometric integrals.** For integrals over a full period, the substitution \( z = e^{i\theta} \) converts \( \int_0^{2\pi} \) into an integral around the unit circle, with \( \cos\theta = \frac{1}{2}(z + 1/z) \) and \( \sin\theta = \frac{1}{2i}(z - 1/z) \). For \( a > 1 \), this converts \( \int_0^{2\pi} \frac{d\theta}{a + \cos\theta} \) into a rational contour integral whose only enclosed pole is inside the unit circle, giving \( \frac{2\pi}{\sqrt{a^2 - 1}} \).<sup>[10](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_22.html)</sup>

**Branch cuts and keyhole contours.** Integrands with non-integer powers or logarithms have branch points, and no closed contour may cross a branch cut. The keyhole contour runs along both sides of the cut, around a large circle, and back around a small circle about the branch point. The two straight pieces are traversed in opposite directions, but the multi-valued factor differs between them by the phase \( e^{2\pi i a} \), so they add rather than cancel. For \( f(z) = z^a/(1+z)^2 \) with \( -1 < a < 1 \) and the cut along the positive real axis, the double pole at \( z = -1 \) is the only enclosed singularity, and the contour calculation yields \( \int_0^{\infty} \frac{x^a}{(1+x)^2}\,dx = \frac{\pi a}{\sin(\pi a)} \).<sup>[10](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_22.html)</sup>

**Poles on the contour.** The theorem as stated requires \( f \) to be analytic on \( C \) itself; a pole lying on the contour voids it.<sup>[11](https://ondara.net/course/complex-analysis/subunit/u6-6-indented-keyhole-contours)</sup>

## Related results and applications

The argument principle follows from the residue theorem applied to \( f'/f \): for a meromorphic function (one whose singularities are at worst poles) and a positively oriented simple closed contour avoiding its zeros and poles, the winding number of the image curve around 0 equals the number of zeros minus the number of poles inside, counted with multiplicity. In control theory this becomes the Nyquist criterion, which counts encirclements of \( -1 \) by the [Nyquist plot](https://www.edgepedia.org/nyquist-plot) to determine closed-loop stability.<sup>[12](https://courses.physics.illinois.edu/ece486/fa2026/documentation/handbook/lec18.html)</sup>

Beyond definite integrals, residue calculus is used to sum certain infinite series, to compute inverse Laplace transforms through the Bromwich inversion formula, and, via Fourier and [Laplace transform](https://www.edgepedia.org/laplace-transform) techniques, to construct solutions of certain partial differential equations.<sup>[2](https://www.cambridge.org/highereducation/books/introduction-to-complex-variables-and-applications/34E6AB639B24E9193860CBBA32AB72F8/residue-calculus-and-applications-of-contour-integration/BF299B83D54B571567F9478B9A1AB569)</sup><sup> • </sup><sup>[13](https://www.cambridge.org/highereducation/books/mathematical-methods-for-physics-and-engineering/FC466374D5B94E86D969100070CA6483/applications-of-complex-variables/441EF9DFD165DA0534C540E8C67B05F7)</sup>

## History

[Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy) introduced the notion of the residue of a function at a point where the function becomes infinite in his paper of 1826, the first number of the *Exercices de Mathématiques*, and devoted many papers in the *Exercices* of 1826 to 1827 to what he called the calculus of residues.<sup>[5](https://www.cambridge.org/core/books/abs/cauchy-and-the-creation-of-complex-function-theory/calculus-of-residues/47BA5BADA5CCDA7531A3008F15EE6A70)</sup> The residue result with respect to poles grew out of his 1825 mémoire, in which he allowed a changing path to cross a simple polar singularity and interpreted the integral by its principal value; the extension to general isolated singularities was made by P. A. Laurent in 1843.<sup>[6](https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf)</sup> The historian of mathematics [Hans Freudenthal](https://www.edgepedia.org/hans-freudenthal), in his [Dictionary of Scientific Biography](https://www.edgepedia.org/dictionary-of-scientific-biography) article on Cauchy, observed that by allowing simple singularities on the integration path, Cauchy handled his residue theorem as a more powerful tool than the narrower formulation found in modern textbooks.<sup>[6](https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf)</sup>

## References

1. [Section 8.1: The Residue Theorem, complexanalysis.org](https://complexanalysis.org/web/sec_residue-thm.html)
2. [Chapter 4: Residue Calculus and Applications of Contour Integration, Ablowitz & Fokas, *Introduction to Complex Variables and Applications*, Cambridge University Press](https://www.cambridge.org/highereducation/books/introduction-to-complex-variables-and-applications/34E6AB639B24E9193860CBBA32AB72F8/residue-calculus-and-applications-of-contour-integration/BF299B83D54B571567F9478B9A1AB569)
3. [Lecture 21: The Residue Theorem, UC Berkeley Complex Analysis notes](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_21.html)
4. [Notes 16: Residues, MATH2230A, The Chinese University of Hong Kong](https://www.math.cuhk.edu.hk/course_builder/1516/math2230a/Notes16.pdf)
5. [Frank Smithies, "The Calculus of Residues," in *Cauchy and the Creation of Complex Function Theory*, Cambridge University Press](https://www.cambridge.org/core/books/abs/cauchy-and-the-creation-of-complex-function-theory/calculus-of-residues/47BA5BADA5CCDA7531A3008F15EE6A70)
6. [Hans Freudenthal, "Augustin-Louis Cauchy," Dictionary of Scientific Biography, University of St Andrews](https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf)
7. [Section 5.3: Residue Theorem, Ximera (Ohio State University)](https://ximera.osu.edu/math/complexBook/complexBook/complexResidue/complexResidue)
8. [Section 9.2: Cauchy's Residue Theorem, Donald Robertson, University of Manchester](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/22-23/29142/residues.html)
9. [Residue Theorem: Contour Integrals via Algebra at the Poles, Unseel](https://unseel.com/mathematics/residue-theorem)
10. [Lecture 22: Applications of the Residue Theorem, UC Berkeley Complex Analysis notes](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_22.html)
11. [Cauchy Principal Value, Complex Analysis course notes, ondara.net](https://ondara.net/course/complex-analysis/subunit/u6-6-indented-keyhole-contours)
12. [Lecture 18: The Argument Principle and Nyquist Criterion, ECE 486 Control Systems, University of Illinois](https://courses.physics.illinois.edu/ece486/fa2026/documentation/handbook/lec18.html)
13. [Chapter 25: Applications of Complex Variables, Riley, Hobson & Bence, *Mathematical Methods for Physics and Engineering*, Cambridge University Press](https://www.cambridge.org/highereducation/books/mathematical-methods-for-physics-and-engineering/FC466374D5B94E86D969100070CA6483/applications-of-complex-variables/441EF9DFD165DA0534C540E8C67B05F7)

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