# References — Chapter ᚎ(Z): The roundabout *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-16, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch16.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 16.1 {#r-16.1} A contour integral is an integral along an oriented curve; the positive orientation of a circle is counter-clockwise. Cauchy’s 1825 memoir defined integration along arbitrary paths in the complex plane and established path independence in a regular domain. The memoir is the Mémoire sur les intégrales définies prises entre des limites imaginaires; Cauchy reached arbitrary closed paths only in 1846, proving the theorem anew by what is now called Green’s formula, and did not recall the 1825 memoir until 1851. *Status: confirmed* - [MIT 18.783 Elliptic Curves (Fall 2025), Lecture 14: Elliptic curves over C (part I)](https://math.mit.edu/classes/18.783/2025/LectureNotes14.pdf) — defines contour integrals and states that positive orientation of a circle is counter-clockwise. - [Dictionary of Scientific Biography, Augustin-Louis Cauchy](https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf) — describes the 1825 memoir’s definition of integrals over arbitrary complex paths and its path-independence result; its bibliography lists “Cauchy, Mémoire sur les intègrales définies prises entre des limites imaginaires (Paris, 1825),” and “Cauchy finally introduced arbitrary closed integration paths, although not as an immediate consequence of his 1825 mémoire, which he did not remember until 1851 [1, 1st ser., XI, 328]; instead, he proved his integral theorem anew by means of what is now called Green’s formula.” ### 16.2 {#r-16.2} Cauchy’s integral theorem gives zero around a closed curve bounding a region on which the function is holomorphic; because e^z is entire, its unit-circle integral is zero. Cauchy did not write the theorem over an arbitrary closed path until a sequence of Comptes rendus notes of 1846, and he reached it there by what is now called Green’s formula rather than as a consequence of his 1825 memoir, which he did not recall until 1851. Goursat’s 1884 and 1900 papers established the theorem without assuming continuity of the derivative. Before 1846 Cauchy restricted himself to rectangular and circular paths; the first of the 1846 notes is Sur les intégrales qui s’étendent à tous les points d’une courbe fermée (C. R. t. XXIII, 3 August 1846). *Status: confirmed* - [Encyclopedia of Mathematics, Analytic function](https://encyclopediaofmath.org/wiki/Holomorphic_function) — states Cauchy’s theorem with the required domain condition and the equivalence of holomorphy and complex analyticity. - [MacTutor, Édouard Goursat](https://mathshistory.st-andrews.ac.uk/Biographies/Goursat/) — identifies Goursat’s removal of the derivative-continuity condition and dates his first paper to 1884. - [Dictionary of Scientific Biography, Édouard Goursat](https://mathshistory.st-andrews.ac.uk/DSB/Goursat.pdf) — records the 1884 Acta Mathematica paper and the corresponding 1900 Transactions of the AMS paper. - [Œuvres complètes d’Augustin Cauchy, 1st series, vol. X, table of contents (Mathdoc)](http://sites.mathdoc.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_10) — lists “Sur les intégrales qui s’étendent à tous les points d’une courbe fermée p.70-74 C. R., t. XXIII, p. 251 (3 août 1846).” ### 16.3 {#r-16.3} The residue theorem makes a positively oriented contour integral 2πi times the sum of enclosed residues; applying it to 1/z on the unit circle gives 2πi. Each positive circuit of the origin adds 2πi to the logarithm. *Status: confirmed* - [NIST DLMF §1.10(iv), Residue Theorem](https://dlmf.nist.gov/1.10.iv) — equation 1.10.8 states the residue theorem with the 2πi normalization and positive orientation. - [NIST DLMF §4.2(i), Logarithms](https://dlmf.nist.gov/4.2.i) — “the general logarithm is given by Ln z = ln z + 2kπi, where k is the excess of the number of times the path … crosses the negative real axis in the positive sense over the number of times in the negative sense.” ### 16.4 {#r-16.4} A residue is the coefficient of (z−a)^−1 in a Laurent expansion. The expansion of e^(1/z) gives residue 1 and an essential singularity at zero; Great Picard’s theorem says every complex value, with at most one exception, occurs infinitely often in every neighborhood of such a point. Since exp has no zeros, e^(1/z) omits the value 0. The residue theorem: (1/2πi)∫_C f(z) dz is the sum of the residues within C. *Status: confirmed* - [NIST DLMF §1.10(iii), Laurent Series](https://dlmf.nist.gov/1.10.iii) — defines residues as Laurent coefficients, classifies essential singularities (“if aₙ ≠ 0 for infinitely many negative n, then z₀ is an isolated essential singularity”); the exponential series used after substituting 1/z is DLMF 4.2.19, on the §4.2(iii) line. - [Encyclopedia of Mathematics, Essential singular point](https://encyclopediaofmath.org/wiki/Essential_singular_point) — states the infinitely-often form of Great Picard’s theorem with at most one exceptional finite value. - [NIST DLMF §4.2(iii), The exponential function](https://dlmf.nist.gov/4.2.iii) — 4.2.19: exp z = 1 + z/1! + z²/2! + z³/3! + ⋯; “The function exp is an entire function of z, with no real or complex zeros.” - [NIST DLMF §1.10(iv), Residue theorem](https://dlmf.nist.gov/1.10.iv) — “(1/2πi)∫_C f(z) dz = sum of the residues of f(z) within C.” ### 16.5 {#r-16.5} The Cauchy–Riemann equations are equivalent, in Wirtinger notation, to ∂f/∂z̄ = 0. Wirtinger introduced the operators in a ~1926 paper; the history and the publisher’s record date it differently and both are given below. *Status: confirmed* - [Encyclopedia of Mathematics, Cauchy–Riemann equations](https://encyclopediaofmath.org/wiki/Cauchy-Riemann_equations) — gives both real equations and their ∂/∂z̄ form. - [R. Michael Range, Some landmarks in the history of the tangential Cauchy–Riemann equations](https://www.esi.ac.at/preprints/esi2225.pdf) — traces the operators to Wirtinger’s 1926 paper and displays their definitions and the holomorphy condition — supports 1926. - [W. Wirtinger, Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen, Mathematische Annalen 97 (1927), 357–375 — Springer’s record for the printed volume](https://doi.org/10.1007/BF01447872) — supports 1927 — the publisher’s own record dates the printed volume to 1927. ### 16.6 {#r-16.6} The winding number n(γ, a) = (1/(2πi))∮γ dz/(z−a) is an integer counting signed turns; the counter-clockwise unit circle has winding number 1. Counterclockwise turns count as positive; the residue theorem counts each residue as many times as the curve winds round it; deforming the contour without crossing a singularity leaves the integral unchanged. *Status: confirmed* - [MIT 18.783 Elliptic Curves (Spring 2013), Lecture 17](https://math.mit.edu/classes/18.783/2013/LectureNotes17.pdf) — gives the complex-integral formula, calls it an integer, and identifies it as the number of turns around the point. - [NIST DLMF §1.10(iv), Residue Theorem](https://dlmf.nist.gov/1.10.iv) — supplies the 1/(2πi) normalization and the residue theorem for a simple closed contour (1.10.8), where “the path C is described in the positive sense”; the winding-number weighting of residues is on the Residue theorem line below. - [Winding number, Wikipedia](https://en.wikipedia.org/wiki/Winding_number) — read as raw wikitext: “When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative.” - [Residue theorem, Wikipedia](https://en.wikipedia.org/wiki/Residue_theorem) — read as raw wikitext: “the line integral of f around γ is equal to 2πi times the sum of residues, each counted as many times as γ winds around the respective point.” - [Contour integration, Wikipedia](https://en.wikipedia.org/wiki/Contour_integration) — read as raw wikitext: “the integrals of holmorphic functions are invariant under deforming the contour, provided the deformation does not cross a singularity or branch cut.” ### 16.7 {#r-16.7} The argument principle gives (1/(2πi))∮ f’/f dz = N−P around a positively oriented contour, counting zeros and poles with multiplicity. For z²+1 on |z|=2, the roots ±i lie inside and there are no poles, so the count is 2; winding-number root counting also yields a proof of the fundamental theorem of algebra. *Status: confirmed* - [NIST DLMF §1.10(iv), Phase Principle](https://dlmf.nist.gov/1.10.iv) — equation 1.10.9 states the argument principle as N−P and specifies multiplicities. - [MIT 18.095, The notion of topological degree](https://math.mit.edu/classes/18.095/2021IAP/) — the course record identifies winding number as the method used for a proof of the fundamental theorem of algebra. ### 16.8 {#r-16.8} A holomorphic function is complex-differentiable on an open set and is locally equal to its Taylor series; a meromorphic function is holomorphic except at isolated poles. *Status: confirmed* - [Encyclopedia of Mathematics, Analytic function](https://encyclopediaofmath.org/wiki/Holomorphic_function) — states the equivalence of holomorphy and complex analyticity and gives the Cauchy integral framework. - [NIST DLMF §1.10(iii), Laurent Series](https://dlmf.nist.gov/1.10.iii) — defines meromorphic functions through isolated poles and analytic functions through Taylor expansions. ### 16.9 {#r-16.9} A Laurent series is a two-sided power series on an annulus and its a_−1 coefficient is the residue. Laurent submitted the result in 1843; Weierstrass had proved it in 1841 but did not publish it then. *Status: confirmed* - [NIST DLMF §1.10(iii), Laurent Series](https://dlmf.nist.gov/1.10.iii) — states the annular two-sided expansion and identifies the residue coefficient. - [MacTutor, Pierre Alphonse Laurent](https://mathshistory.st-andrews.ac.uk/Biographies/Laurent_Pierre/) — dates Laurent’s submission and Cauchy’s report to 1843 and documents Weierstrass’s unpublished 1841 proof: “He submitted a memoir for the Grand Prix of the Academy of Sciences of 1842, unfortunately after the final date for submission in 1843”; “Cauchy reported on Laurent’s entry Mémoire sur le calcul des variations, which contains the Laurent series for a complex function, on 20 May 1843.” ### 16.10 {#r-16.10} The residue theorem reduces contour integrals to residue sums and supports the evaluation of real integrals. Cauchy began the calculus of residues in 1826, in Sur un nouveau genre de calcul analogue au calcul infinitésimal, which is where the word residue enters mathematics; his Turin memoirs date to 1831–32, were summarized in 1837, and were republished in 1840–41. Boas notes that only in 1826 does Cauchy introduce the word residue. Contour integration gives ∫e^(itx)/(x² + 1) dx = πe^(−t) for t ≥ 0, which is π at t = 0. *Status: confirmed* - [Dictionary of Scientific Biography, Augustin-Louis Cauchy](https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf) — documents the Turin memoirs, their later publication history, and Cauchy’s use of residues for definite integrals. - [NIST DLMF §1.10(iv), Residue Theorem](https://dlmf.nist.gov/1.10.iv) — gives the modern theorem that converts a contour integral to a finite residue sum: “If f(z) is analytic within a simple closed contour C, and continuous within and on C—except in both instances for a finite number of singularities within C—then” (1/2πi)∫_C f(z) dz is the sum of the residues within C (1.10.8). - [Augustin-Louis Cauchy, MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/Cauchy/) — “He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal.” - [Harold P. Boas, Cauchy’s residue sore thumb, arXiv:1701.04887](https://arxiv.org/abs/1701.04887) — read from the PDF: “Cauchy does not have the present terminology of line integrals (path integrals), and only in 1826 does he introduce the word “residue.”” - [Contour integration, Wikipedia](https://en.wikipedia.org/wiki/Contour_integration) — read as raw wikitext: the worked example ∫ e^(itx)/(x² + 1) dx over the real line = πe^(−t) (t ≥ 0). ### 16.11 {#r-16.11} A bounded entire function is constant. Cauchy apparently first published the result in 1844 and Liouville presented it in lectures in 1847; it gives a standard short proof of the fundamental theorem of algebra. *Status: confirmed* - [Encyclopedia of Mathematics, Liouville theorems](https://encyclopediaofmath.org/wiki/Liouville_theorems) — states the theorem and records the Cauchy 1844 and Liouville 1847 history. - [MIT 18.04, Complex Variables with Applications](https://math.mit.edu/~dunkel/Teach/18.04_2019S/notes/1804_Main.pdf) — states Liouville’s theorem and derives the fundamental theorem of algebra from 1/p. ### 16.12 {#r-16.12} Riemann’s 1851 thesis developed complex-function theory using what are now called Riemann surfaces and conformal mappings; such surfaces make multivalued relations single-valued, and compact surfaces are classified in part by genus. √z has two branches and log z needs countably many sheets; a Riemann surface is a connected one-dimensional complex manifold, and the Riemann surfaces on the torus are the elliptic curves. *Status: confirmed* - [MacTutor, Bernhard Riemann](https://mathshistory.st-andrews.ac.uk/Biographies/Riemann/) — dates the thesis to 1851 and describes its Riemann surfaces, conformal mappings, and connectivity. - [Encyclopedia of Mathematics, Riemann surface](https://encyclopediaofmath.org/wiki/Riemann_surface) — explains single-valued uniformization, the 1851 result, and classification data by genus: “Closed Riemann surfaces are topologically completely determined by one number — the genus.” - [Branch point, Wikipedia](https://en.wikipedia.org/wiki/Branch_point) — read as raw wikitext: “the function w = z^(1/2) has two branches”; “The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut.” - [Riemann surface, Wikipedia](https://en.wikipedia.org/wiki/Riemann_surface) — read as raw wikitext: “a Riemann surface is a connected one-dimensional complex manifold”; “The 2-torus T² has many different Riemann surface structures, all of the form ℂ/(ℤ+τℤ) … These are called elliptic curves.” ### 16.13 {#r-16.13} The holomorphic automorphisms of the Riemann sphere are Möbius transformations forming PSL(2, ℂ), and they preserve generalized circles. PSL(2, ℂ) corresponds to the connected, proper orthochronous Lorentz group, not the full Lorentz group. *Status: confirmed* - [Harvard Math 55b course notes](https://people.math.harvard.edu/~ctm/home/text/class/harvard/55b/10/html/home/course/course.pdf) — states Aut(Ĉ) ≅ PSL₂(ℂ) and that these automorphisms send circles to circles. - [Blagoje Oblak, From the Lorentz Group to the Celestial Sphere](https://arxiv.org/abs/1508.00920) — the published lecture notes identify the connected Lorentz group with the conformal transformations of the sphere. ### 16.14 {#r-16.14} A holomorphic map with nonzero derivative is conformal, and every simply connected proper plane domain is biholomorphic to the disc. Riemann formulated the mapping theorem in 1851; Mercator’s 1569 world map uses a conformal projection. *Status: confirmed* - [Encyclopedia of Mathematics, Conformal mapping](https://encyclopediaofmath.org/wiki/Conformal_mapping) — defines conformality and connects the simply connected case to the Riemann mapping theorem. - [Encyclopedia of Mathematics, Riemann theorem](https://encyclopediaofmath.org/wiki/Riemann_theorem) — states the mapping theorem and dates Riemann’s formulation to 1851. - [Cartographica Helvetica, The maps of Europe by Gerard Mercator](https://www.kartengeschichte.ch/ch/summaries/e10a.html) — dates the world map to 1569 and identifies its cylindrical projection as conformal. ### 16.15 {#r-16.15} Hadamard printed the epigraph’s English sentence in his 1945 book; the underlying thought is adapted from a related sentence in Paul Painlevé’s 1900 account of his scientific work. *Status: confirmed* - [Jacques Hadamard, An Essay on the Psychology of Invention in the Mathematical Field](https://archive.org/details/eassayonthepsych006281mbp) — the 1945 scan contains the quoted wording on page 123. - [University of Iowa, source history for the Hadamard quotation](https://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html) — transcribes Hadamard’s sentence and traces the related original wording to Painlevé’s 1900 Analyse des travaux scientifiques. ### 16.16 {#r-16.16} Cauchy’s integral formula: if D is an open domain whose boundary consists of finitely many non-intersecting rectifiable curves oriented positively with respect to D, and f is holomorphic on D, then f(z) = (1/2πi)∫_∂D f(ζ)/(ζ − z) dζ for every z in D, so the values of a holomorphic function inside a loop are determined by its values on the loop. The formula makes a holomorphic function infinitely differentiable and, through the geometric series, equal to its power series; Cauchy’s integral theorem yields the formula. *Status: confirmed* - [Encyclopedia of Mathematics, Analytic function](https://encyclopediaofmath.org/wiki/Holomorphic_function) — states the formula in so many words — “if $D$ is an open domain whose boundary consists of a finite number of non-intersecting rectifiable curves (oriented positively with respect to $D$) and $f: D\to \mathbb C$ is holomorphic, then $f(z) = \frac{1}{2\pi i} \int_{\partial D} \frac{f (\zeta)}{\zeta-z}\, d\zeta \quad \forall z\in D$” — and carries the hypotheses (open domain, finitely many non-intersecting rectifiable boundary curves, positive orientation) that the popular one-line version drops. - [Cauchy’s integral formula, Wikipedia](https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula) — read as raw wikitext: “it implies that a function that is holomorphic in an open set is in fact infinitely differentiable there. Furthermore, it is an analytic function, meaning that it can be represented as a power series. The proof of this uses the dominated convergence theorem and the geometric series.” - [Analytic function, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Analytic_function) — “Cauchy’s integral theorem yields Cauchy’s integral formula, which expresses the values of an analytic function inside a domain in terms of its values on the boundary.” Editorial note: Added in the V14 prose pass for the new card `cauchyformula`. The chapter’s sentence “One complex derivative forces infinitely many, equality with the function’s own power series, and complete determination inside a loop by the values on it” is footnoted 16.8, and 16.8’s first source is this same Encyclopedia of Mathematics page, so the third clause was already carried by a source the chapter cites; what 16.8’s claim line states is only the holomorphic/analytic equivalence and the definition of meromorphic, which is why the integral formula is given its own entry here rather than being read into 16.8. ### 16.18 {#r-16.18} States Green’s theorem, converting a line integral around a closed plane curve into a double integral over the enclosed region, from Green’s 1828 essay (a proof was finally given by Riemann in 1851), and its role as the two-dimensional case of Stokes’ theorem underlying Cauchy’s integral theorem. *Status: confirmed* - [Green’s theorem, Wikipedia](https://en.wikipedia.org/wiki/Green%27s_theorem) — states the theorem, dates it to George Green’s 1828 essay (Green “stated a similar result in an 1828 paper titled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism”), says “A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his doctoral dissertation,” and describes its relation to Stokes’ theorem.