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# Glossary cards — Chapter ᚎ(Z): The roundabout

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T16, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch16.md)

*Complex analysis: contour integrals, Cauchy’s theorem, the Cauchy–Riemann equations, and residues. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Straif — `ᚎ` {#g-og-straif}

*Ogham letter*

Ogham letter name meaning ‘sulphur’; the medieval glossators read it as the blackthorn.

Straif, Old Irish Straiph, means ‘sulphur’. Its Primitive Irish sound value is uncertain: it may have been a sibilant distinct from the s that sail already held, perhaps a reflex of /st/ or /sw/. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give tressam rúamnai, “strongest reddening, that is, dye”; mórad rún, “increase of secrets”; and saigid nél, “seeking of clouds.” The Auraicept glosses draighin, blackthorn: “the hedge of a stream is sraibh.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19)

### contour integral around the unit circle — `∮_{|z|=1}` {#g-contour}

*Function or operator*

Integrate once counterclockwise around the circle of radius one.

The circle symbol on the integral sign means a path integral around a closed curve, here the circle of radius one centered at the origin, traversed once counterclockwise. Cauchy defined integrals along paths in the complex plane in his memoir of 1825. An operator, so uncolored.

Cites: [16.1](ch16-refs.md#r-16.1)

See also: [the complex plane](ch21-cards.md#g-cplane)

### Cauchy’s integral theorem — `∮ e^{z} dz` {#g-cauchy}

*Integer or rational*

The integral of any entire function around any closed loop is 0 (Cauchy: path independence 1825, arbitrary closed loops 1846).

Cauchy’s integral theorem says that the integral of a holomorphic function around a closed loop is 0 when the loop and its interior lie in the function’s domain. Cauchy proved path-independence in his 1825 memoir Sur les intégrales définies, prises entre des limites imaginaires, but did not write the theorem over an arbitrary closed curve until the 1846 note Sur les intégrales qui s’étendent à tous les points d’une courbe fermée; the closed-loop form follows from the 1825 result as mathematics, though not as history: Cauchy proved it afresh by what is now called Green’s formula, and did not recall the 1825 memoir until 1851. Goursat’s 1884 and 1900 papers removed Cauchy’s extra continuity assumption on the derivative. Since e^z is entire, its integral around the unit circle is exactly 0, an integer.

Cites: [16.1](ch16-refs.md#r-16.1), [16.2](ch16-refs.md#r-16.2)

See also: [Green’s theorem](ch16-cards.md#g-greenthm), [holomorphic and meromorphic](ch16-cards.md#g-holomorphic)

### the integral that refuses to vanish — `∮ dz/z` {#g-dzz}

*Non-real object*

Equal to 2πi once counterclockwise around the origin, the source of every residue.

∮ dz/z around the counterclockwise unit circle equals 2πi, because 1/z has a pole at the origin inside the loop and log z gains 2πi on each trip around it. It is the one integral every residue traces back to, and its value is a non-real complex number, which is why it carries the color of a non-real object. With the residue, the winding number, and the root count beside it, this chapter’s denominator is 2πi + 4, whose real part is 4, so it is certainly not zero.

Cites: [16.3](ch16-refs.md#r-16.3)

See also: [the argument principle, counted](ch16-cards.md#g-argprin), [residue at an essential singularity](ch16-cards.md#g-res), [the winding number of the unit circle about the origin](ch16-cards.md#g-winding)

### residue at an essential singularity — `Res_{0} e^{1/z}` {#g-res}

*Integer or rational*

Equal to 1, read off the Laurent series of e^(1/z).

The residue of e^(1/z) at 0 is the coefficient of 1/z in its Laurent expansion 1 + 1/z + 1/(2!z²) + …, which is 1. The point is an essential singularity; Great Picard’s theorem says that, with at most one exception, every complex value occurs infinitely often in every neighborhood of such a point. Here the omitted value is 0, yet the residue is a plain integer.

Cites: [16.4](ch16-refs.md#r-16.4)

See also: [the Laurent series](ch16-cards.md#g-laurent), [Great Picard’s theorem](ch16-cards.md#g-picard)

### the Cauchy–Riemann equations in one line — `∂f/∂z̄` {#g-wirtinger}

*Integer or rational*

Exactly 0 for every holomorphic f: this one equation is the Cauchy–Riemann test for complex differentiability.

The Cauchy–Riemann condition for complex differentiability is the pair of real equations u_{x} = v_{y} and u_{y} = −v_{x}, where f = u + iv. Wirtinger’s operators, introduced in a ~1926 paper, fold them into ∂f/∂z̄ = ½(∂_{x} + i∂_{y})f = 0. The tilde is there because Range’s history calls it the 1926 paper while Springer’s own record dates the printed volume, Math. Ann. 97, 357–375, to 1927; both are cited at 16.5. It is the condition behind the chapter’s holomorphic calculations. Exactly 0, an integer.

Cites: [16.5](ch16-refs.md#r-16.5)

See also: [holomorphic and meromorphic](ch16-cards.md#g-holomorphic)

### the residue of an entire function — `Res_{0} e^{z}` {#g-resentire}

*Integer or rational*

Exactly 0: a function with no singularity has no 1/z term in its expansion, so its residue anywhere is 0.

The residue at a point is the coefficient of 1/(z − a) in the Laurent series there; for a function holomorphic at a the series has no negative powers, so the residue is 0. That is the residue theorem’s way of restating Cauchy’s theorem: the integral around a loop is 2πi times the residues inside, and an entire function has none. Exactly 0, an integer.

Cites: [16.4](ch16-refs.md#r-16.4)

See also: [Cauchy’s integral theorem](ch16-cards.md#g-cauchy), [holomorphic and meromorphic](ch16-cards.md#g-holomorphic), [the Laurent series](ch16-cards.md#g-laurent), [the residue theorem](ch16-cards.md#g-residuethm)

### the winding number of the unit circle about the origin — `n(γ, 0)` {#g-winding}

*Integer or rational*

Equal to 1: the counterclockwise circle goes round the origin once, which is exactly why ∮ dz/z is 2πi and not 0.

The winding number n(γ, a) = (1/2πi)∮_γ dz/(z − a) is an integer counting the signed turns of a closed curve γ around a; for the counterclockwise unit circle about the origin it is 1. The residue theorem weights each enclosed residue by this count, so deformations that do not cross a singularity leave the integral unchanged. An integer.

Cites: [16.3](ch16-refs.md#r-16.3), [16.6](ch16-refs.md#r-16.6)

See also: [the integral that refuses to vanish](ch16-cards.md#g-dzz), [the residue theorem](ch16-cards.md#g-residuethm)

### the argument principle, counted — `#{z : z^{2} + 1 = 0, |z| < 2}` {#g-argprin}

*Integer or rational*

Equal to 2: (1/2πi)∮ f′/f dz counterclockwise around |z| = 2 counts the zeros of f = z² + 1 inside, which are i and −i.

For f meromorphic inside a positively oriented loop and nonzero on it, the argument principle says that (1/2πi)∮ f′(z)/f(z) dz equals the number of zeros minus the number of poles inside, counted with multiplicity. The polynomial z² + 1 has the two roots ±i and no poles, all inside |z| = 2, so the count is 2. Root-counting by winding number also gives a proof of the fundamental theorem of algebra. An integer.

Cites: [16.7](ch16-refs.md#r-16.7)

See also: [the fundamental theorem of algebra, counted](ch21-cards.md#g-fta)

### holomorphic and meromorphic — `f ∈ 𝒪(U)` {#g-holomorphic}

*Function or operator*

Holomorphic: complex-differentiable at every point of an open set; meromorphic: holomorphic except for poles. Holomorphic and complex analytic mean the same thing.

A holomorphic function is complex-differentiable on an open set U, which by Cauchy’s integral formula makes it infinitely differentiable and locally equal to its own Taylor series: holomorphic and complex analytic are equivalent. A meromorphic function is holomorphic except at isolated poles. A property, uncolored.

Cites: [16.8](ch16-refs.md#r-16.8)

See also: [Cauchy’s integral formula](ch16-cards.md#g-cauchyformula)

### the Laurent series — `Σ_{n∈ℤ} a_{n}(z − a)^{n}` {#g-laurent}

*Function or operator*

A power series allowed negative powers, valid on an annulus; the coefficient a₋₁ is the residue.

Pierre Alphonse Laurent submitted the expansion to the Academy of Sciences in 1843; Weierstrass had proved the result in 1841 but did not publish it then. On any annulus where f is holomorphic, f has a two-sided power series in z − a, and the coefficient of 1/(z − a) is its residue. A tool, uncolored.

Cites: [16.4](ch16-refs.md#r-16.4), [16.9](ch16-refs.md#r-16.9)

See also: [holomorphic and meromorphic](ch16-cards.md#g-holomorphic)

### the residue theorem — `∮_{γ} f dz = 2πi Σ Res f` {#g-residuethm}

*Function or operator*

Around a positively oriented simple closed curve, with the function holomorphic inside except at finitely many isolated singularities and none on the curve, the integral is 2πi times the sum of the residues inside: Cauchy’s theorem with the singularities put back.

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 of 1831–32 were summarized in 1837 and republished in 1840–41. The theorem turns contour integrals into algebraic sums and evaluates real integrals such as ∫ dx/(1 + x²) = π over the whole real line. Every residue traces back to ∮ dz/z = 2πi in this chapter’s denominator. A theorem, uncolored.

Cites: [16.10](ch16-refs.md#r-16.10), [16.3](ch16-refs.md#r-16.3)

See also: [Cauchy’s integral theorem](ch16-cards.md#g-cauchy), [the integral that refuses to vanish](ch16-cards.md#g-dzz), [holomorphic and meromorphic](ch16-cards.md#g-holomorphic)

### Liouville’s theorem — `bounded and entire ⇒ constant` {#g-liouville}

*Function or operator*

A bounded holomorphic function on the whole plane is constant; Cauchy apparently first published it in 1844, Liouville got the name.

Cauchy apparently first published the bounded-entire theorem in 1844; Liouville presented it in lectures in 1847, which is how his name became attached to it. It gives a short proof of the fundamental theorem of algebra: if a nonconstant polynomial p had no root, 1/p would be bounded and entire. Chapter ᚔ(I) counts those roots. A theorem, uncolored.

Cites: [16.11](ch16-refs.md#r-16.11)

See also: [the fundamental theorem of algebra, counted](ch21-cards.md#g-fta), [holomorphic and meromorphic](ch16-cards.md#g-holomorphic)

### a Riemann surface — `Riemann surface` {#g-riemannsurf}

*Non-real object*

The natural home of a multivalued function such as √z or log z: sheets glued along cuts so that the function becomes single-valued.

Riemann introduced them in his 1851 thesis to make sense of functions like √z, which needs two sheets, and log z, which needs infinitely many; they are one-dimensional complex manifolds, and the compact ones are classified topologically by genus, which ties this chapter to the topology of Chapter ᚊ(Q) and to the elliptic curves of Chapter ᚒ(U), which are Riemann surfaces of genus one. An object, so colored as a non-real object.

Cites: [16.12](ch16-refs.md#r-16.12)

See also: [the Weierstrass form](ch19-cards.md#g-weierstrass)

### a Möbius transformation — `z ↦ (az + b)/(cz + d)` {#g-mobiustrans}

*Function or operator*

The invertible holomorphic maps of the Riemann sphere that send circles to circles; the group of them is PSL(2, ℂ).

These fractional linear maps are exactly the holomorphic automorphisms of the Riemann sphere. They send generalized circles—ordinary circles or lines—to generalized circles and form PSL(2, ℂ), which is isomorphic to the proper orthochronous Lorentz group of special relativity. A map, uncolored.

Cites: [16.13](ch16-refs.md#r-16.13)

See also: [holomorphic and meromorphic](ch16-cards.md#g-holomorphic), [the Riemann sphere](ch21-cards.md#g-riemannsphere)

### conformal maps — `f′(z) ≠ 0 ⇒ angles preserved` {#g-conformal}

*Function or operator*

Wherever a holomorphic function has nonzero derivative it preserves angles; Riemann’s mapping theorem says any simply connected region but the plane can be mapped conformally and one-to-one onto the disc.

A holomorphic map with f′(z) ≠ 0 multiplies infinitesimal figures by the complex number f′(z), a rotation and a scaling, so angles survive. Riemann formulated his mapping theorem in 1851; later work supplied rigorous proofs. Mercator’s 1569 world map is a historic conformal projection. A property, uncolored.

Cites: [16.14](ch16-refs.md#r-16.14)

See also: [holomorphic and meromorphic](ch16-cards.md#g-holomorphic), [an infinitesimal](ch26-cards.md#g-infinitesimal)

### Cauchy’s integral formula — `f(z) = (1/2πi)∮ f(ζ)/(ζ − z) dζ` {#g-cauchyformula}

*Function or operator*

A holomorphic function’s values inside a loop are fixed by its values on the loop, and the fixing is an integral.

If f is holomorphic on a domain whose boundary is finitely many non-intersecting rectifiable curves, traversed in the positive sense, then f(z) = (1/2πi)∮ f(ζ)/(ζ − z) dζ at every interior point z. It is the reason holomorphic functions are as rigid as this chapter says: differentiate under the integral sign and one complex derivative becomes infinitely many, and expanding 1/(ζ − z) as a geometric series turns the same integral into the function’s own Taylor series. Cauchy’s theorem, from which the formula is derived, says that with nothing singular inside, a holomorphic function’s integral around the loop is 0, which is this chapter’s numerator. A theorem, uncolored.

Cites: [16.16](ch16-refs.md#r-16.16)

See also: [Cauchy’s integral theorem](ch16-cards.md#g-cauchy), [holomorphic and meromorphic](ch16-cards.md#g-holomorphic)

### Great Picard’s theorem — `essential singularity ⇒ every value but one` {#g-picard}

*Function or operator*

In every neighborhood of an essential singularity a function takes every finite complex value, with at most one possible exception, infinitely often.

If f is holomorphic in a punctured neighborhood of an essential singular point, then in every neighborhood of that point, however small, each finite complex value with at most one possible exception is taken infinitely often. For e^(1/z) at the origin the one exception is 0, because the exponential is never 0; every other complex number is hit infinitely often inside any disc round the origin. The chapter puts it beside a residue that is nevertheless a plain 1. A theorem, uncolored.

Cites: [16.4](ch16-refs.md#r-16.4)

See also: [holomorphic and meromorphic](ch16-cards.md#g-holomorphic), [residue at an essential singularity](ch16-cards.md#g-res)

### Green’s theorem — `∮(Ldx+Mdy)=∬(∂M/∂x−∂L/∂y)dA` {#g-greenthm}

*Function or operator*

Green’s theorem converts a line integral around a closed curve in the plane into a double integral over the region the curve encloses, relating circulation on the boundary to a derivative inside.

Named for George Green, who stated a version of it in an 1828 essay on electricity and magnetism, though a proof came only with Riemann in 1851, it says ∮(L dx + M dy) = ∬(∂M/∂x − ∂L/∂y) dA around a positively oriented simple closed curve. It is the two-dimensional case of the more general Stokes’ theorem, and in complex analysis it underlies Cauchy’s integral theorem, the statement that a holomorphic function’s integral around a closed loop vanishes. Applied to L = −y/2, M = x/2 it gives a way to compute a region’s area purely from its boundary.

Cites: [16.18](ch16-refs.md#r-16.18)

See also: [Cauchy’s integral theorem](ch16-cards.md#g-cauchy), [holomorphic and meromorphic](ch16-cards.md#g-holomorphic), [Stokes’ theorem](ch11-cards.md#g-stokes)