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# Glossary cards — Chapter ᚐ(A): The mechanic’s workshop

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T17, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch17.md)

*Special functions: gamma and its family, the logarithmic integral, sinc, Bessel, Lambert W, the error function, Airy, the elliptic integral, and Fresnel. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Ailm — `ᚐ` {#g-og-ailm}

*Ogham letter*

Ogham letter name of uncertain meaning, possibly ‘pine’; its kennings are all about sound.

Ailm is of uncertain meaning. McManus concedes it may mean ‘pine-tree’, since an eighth-century poem appears to use it so—the hermit Marbán saying caine ailmi ardom-peitet, “beautiful are the pines that make music for me,” of the sound of wind in the needles. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give ardam íachta, “loudest groan”; tosach frecrai, “beginning of an answer”; and tosach garmae, “beginning of calling.” The Auraicept glosses crand giuis or ochtach, fir or pine, so the plant column and the philology point the same way here, unusually for a name this uncertain. 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)

### logarithmic integral — `li` {#g-li}

*Function or operator*

The principal-value integral ∫ dt/ln t, the function in the prime number theorem, π(x) ~ li(x).

The logarithmic integral is the principal-value integral li(x) = PV∫₀ˣ dt/ln t across its singularity at t = 1. The prime number theorem can be written π(x) ~ li(x)—the form DLMF sharpens at §27.12 into a bound on |π(x) − li(x)|. A function, not a number, so uncolored.

Cites: [17.4](ch17-refs.md#r-17.4)

See also: [the prime number theorem](ch18-cards.md#g-pnt), [the prime-counting function](ch18-cards.md#g-primecount)

### Ramanujan–Soldner constant — `μ` {#g-mu}

*Status open*

The unique positive zero of li, about 1.45137; no cited source settles its irrationality.

The Ramanujan–Soldner constant, μ ≈ 1.4513692349, is the unique positive zero of li. Soldner tabulated the logarithmic integral in 1809, and Ramanujan later computed the constant. No source cited here settles whether μ is irrational, so it remains status open; li(μ) = 0 is its definition, which is why it can be a numerator.

Cites: [17.5](ch17-refs.md#r-17.5)

See also: [logarithmic integral](ch17-cards.md#g-li)

### gamma of one half — `Γ(½)` {#g-gammahalf}

*Proven transcendental*

Equal to √π.

Γ(½) = √π ≈ 1.7725, and the recurrence Γ(z + 1) = zΓ(z) makes Γ(½) the value (−½)! while (½)! = Γ(3/2) = √π/2. It is transcendental because π is transcendental.

Cites: [17.12](ch17-refs.md#r-17.12), [17.3](ch17-refs.md#r-17.3)

See also: [pi](ch04-cards.md#g-pi)

### the omega constant — `W(1)` {#g-W1}

*Proven transcendental*

The solution of x·e^x = 1, about 0.5671, from Lambert’s W function; transcendental.

The omega constant Ω = W(1) ≈ 0.5671433 is the positive solution of Ωe^Ω = 1, where W is the multivalued inverse of w ↦ we^w. Corless, Gonnet, Hare, Jeffrey, and Knuth standardized its branches and the name W in 1996. Lindemann–Weierstrass proves Ω transcendental: if Ω were nonzero algebraic, e^Ω = 1/Ω could not also be algebraic. This Ω is Lambert’s, not Chaitin’s halting probability of Chapter ᚃ(F).

Cites: [17.11](ch17-refs.md#r-17.11), [17.12](ch17-refs.md#r-17.12)

See also: [the Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier), [Chaitin’s constant](ch05-cards.md#g-omega)

### first zero of the Bessel function J₀ — `j_{0,1}` {#g-j01}

*Proven transcendental*

About 2.4048, the note a drumhead cannot help playing; transcendental by Siegel’s theorem of 1929.

The first positive zero of J₀ is j₀,₁ ≈ 2.4048256. Zeros of J₀ set the frequencies of a circular drumhead’s axisymmetric modes, and this first zero gives the fundamental one. Siegel’s 1929 theorem implies that every nonzero zero of J₀ is transcendental.

Cites: [17.13](ch17-refs.md#r-17.13)

### error function at one — `erf(1)` {#g-erf1}

*Status open*

About 0.8427, the Gaussian probability within √2 standard deviations; no cited source settles its irrationality.

erf(1) ≈ 0.8427008 equals the probability that a standard normal variable lies within √2 standard deviations of its mean, because erf(x) = (2/√π)∫₀ˣ e^(−t²)dt. A 2023 Siegel–Shidlovsky argument proves the unnormalized integral ∫₀¹e^(−t²)dt transcendental, but does not determine the arithmetic nature of the ratio erf(1); no cited source settles its irrationality, so it remains status open.

Cites: [17.14](ch17-refs.md#r-17.14), [17.23](ch17-refs.md#r-17.23)

See also: [the error function](ch17-cards.md#g-erffn), [the standard normal density](ch13-cards.md#g-normal), [square root of two](ch04-cards.md#g-sqrt2)

### Airy function at zero — `Ai(0)` {#g-ai0}

*Proven transcendental*

About 0.3550, equal to 1/(3^(2/3) Γ(2/3)), transcendental via Chudnovsky’s result on Γ(1/3).

Ai(0) = 1/(3^(2/3)·Γ(⅔)) ≈ 0.3550280. Airy functions solve the Airy differential equation and occur in linear-potential quantum problems. Chudnovsky’s algebraic-independence work in the 1970s makes π and Γ(⅓) algebraically independent; together with Γ(⅓)Γ(⅔) = 2π/√3, this proves Ai(0) transcendental.

Cites: [17.15](ch17-refs.md#r-17.15), [17.24](ch17-refs.md#r-17.24), [17.27](ch17-refs.md#r-17.27)

See also: [the Airy functions](ch17-cards.md#g-airyfn), [Theodorus’ constant](ch04-cards.md#g-sqrt3)

### the reciprocal gamma function at a pole — `1/Γ(0)` {#g-recipgamma}

*Integer or rational*

Γ has a pole at 0, so its reciprocal, which extends smoothly across the poles, is zero there.

The gamma function has simple poles at 0, −1, −2, …, while its reciprocal 1/Γ is entire and has a simple zero at each of those points. Thus (1/Γ)(0) = 0 exactly, hence an integer; the displayed 1/Γ(0) is standard shorthand for that entire reciprocal-gamma value.

Cites: [17.6](ch17-refs.md#r-17.6)

See also: [the gamma function](ch17-cards.md#g-gammafn)

### the sinc function at pi — `sinc(π)` {#g-sincpi}

*Integer or rational*

Exactly 0: unnormalized sinc x = sin x / x, and the sine vanishes at π while the denominator does not.

For the unnormalized convention sinc(x) = sin x/x, continuity gives sinc(0) = 1 and the nonzero integer multiples of π are its zeros. The sinc kernel also reconstructs a band-limited signal from sufficiently dense samples. Thus sinc(π) = 0 exactly, an integer.

Cites: [17.7](ch17-refs.md#r-17.7)

### the Bessel function of order one at zero — `J_{1}(0)` {#g-j1zero}

*Integer or rational*

Exactly 0: every Bessel function Jₙ with n ≥ 1 starts from nothing, while J₀(0) = 1.

For integer n ≥ 0, the defining series of Jₙ begins with (x/2)ⁿ/n!. Consequently J₀(0) = 1 and Jₙ(0) = 0 for every n ≥ 1, including J₁(0). Exactly 0, an integer.

Cites: [17.8](ch17-refs.md#r-17.8)

See also: [the Bessel functions](ch17-cards.md#g-besselfn)

### the Beta function at one half, one half — `B(½, ½)` {#g-betahalf}

*Proven transcendental*

Equal to π: the Beta function is Γ(a)Γ(b)/Γ(a + b), and here the two Γ(½) give √π·√π.

The beta integral satisfies B(a, b) = Γ(a)Γ(b)/Γ(a+b). Therefore B(½, ½) = Γ(½)²/Γ(1) = π, equivalently the integral of 1/√(t(1−t)) over [0, 1]. It is transcendental because π is.

Cites: [17.12](ch17-refs.md#r-17.12), [17.3](ch17-refs.md#r-17.3), [17.9](ch17-refs.md#r-17.9)

See also: [the beta function](ch17-cards.md#g-betafn), [gamma of one half](ch17-cards.md#g-gammahalf)

### the digamma function at one — `|ψ(1)|` {#g-digamma1}

*Status open*

ψ(1) = −γ: the logarithmic derivative of Γ at 1 is minus the Euler–Mascheroni constant, whose irrationality is unknown.

The digamma function is ψ(x) = Γ′(x)/Γ(x), and its special values satisfy ψ(1) = −γ and ψ(n+1) = Hₙ − γ. Current literature still does not establish whether the Euler–Mascheroni constant γ is irrational, so |ψ(1)| remains status open. It is the same γ as Chapter ᚃ(F)’s.

Cites: [17.10](ch17-refs.md#r-17.10), [17.23](ch17-refs.md#r-17.23)

See also: [the digamma function](ch17-cards.md#g-digammafn), [Euler–Mascheroni constant](ch05-cards.md#g-gamma)

### the other real branch of Lambert W — `|W_{−1}(−1/e)|` {#g-wm1}

*Integer or rational*

Equal to 1: the two real branches of W meet at the branch point (−1/e, −1), so W₋₁(−1/e) = −1.

Lambert W is multivalued because w ↦ we^w is not one-to-one. On the real interval −1/e < x < 0 it has the branches W₀ and W₋₁, and both meet at W(−1/e) = −1. Therefore |W₋₁(−1/e)| = 1, an integer.

Cites: [17.11](ch17-refs.md#r-17.11)

### the complete elliptic integral of the first kind at zero — `K(0)` {#g-K0}

*Proven transcendental*

Equal to π/2: with modulus zero the integral ∫₀^(π/2) dθ/√(1 − k² sin²θ) is just a quarter turn.

The complete elliptic integral of the first kind is K(k) = ∫₀^(π/2) dθ/√(1−k²sin²θ); it gives the finite-amplitude pendulum period. At k = 0 the integrand is 1, so K(0) = π/2, transcendental because π is. The arc length of an ellipse uses a different one, the elliptic integral of the second kind E.

Cites: [17.12](ch17-refs.md#r-17.12), [17.16](ch17-refs.md#r-17.16)

### the Fresnel sine integral at infinity — `S(∞)` {#g-fresnelS}

*Integer or rational*

Equal to ½: ∫₀^∞ sin(πt²/2) dt converges to ½, the point the Cornu spiral winds toward.

The Fresnel integrals are S(x) = ∫₀ˣ sin(πt²/2)dt and C(x) = ∫₀ˣ cos(πt²/2)dt. They parametrically draw the Cornu spiral used in diffraction, and both tend to ½ as x → ∞. Thus S(∞) is exactly the rational number ½.

Cites: [17.17](ch17-refs.md#r-17.17)

See also: [the Cornu spiral](ch17-cards.md#g-cornu), [the Fresnel integrals](ch17-cards.md#g-fresnelfn)

### the Gauss hypergeometric function — `_{2}F_{1}(a, b; c; z)` {#g-hypergeom}

*Function or operator*

The series Σ (a)ₙ(b)ₙ/(c)ₙ · zⁿ/n!, with logarithm, arcsine, and polynomial special cases.

The Gauss hypergeometric function is defined for |z| < 1 by ₂F₁(a, b;c;z) = Σₙ₌₀∞ (a)ₙ(b)ₙzⁿ/((c)ₙn!), with analytic continuation elsewhere. Parameter choices give elementary functions including logarithms and inverse trigonometric functions, while negative integer parameters produce polynomial cases. A function, uncolored.

Cites: [17.18](ch17-refs.md#r-17.18)

See also: [analytic continuation](ch18-cards.md#g-analcont)

### the polylogarithm — `Li_{s}(z)` {#g-polylog}

*Function or operator*

Σ zⁿ/nˢ: Li₁(z) = −ln(1 − z), Liₛ(1) = ζ(s) where the series converges, and Li₂ is the dilogarithm.

The polylogarithm Liₛ(z) = Σₙ₌₁∞ zⁿ/nˢ generalizes the logarithm through Li₁(z) = −ln(1−z) and specializes to the zeta function through Liₛ(1) = ζ(s), where the defining series converges. Its dilogarithm value Li₂(½) = π²/12 − (ln 2)²/2 is a standard special case. A function, uncolored.

Cites: [17.19](ch17-refs.md#r-17.19)

See also: [Riemann zeta function](ch18-cards.md#g-zeta)

### the Jacobi theta function — `ϑ(z; τ)` {#g-thetafn}

*Function or operator*

Σ exp(πin²τ + 2πinz): a building block of elliptic functions, and the function whose transformation law gives the functional equation of ζ.

Jacobi’s theta function ϑ₃(z;τ) = Σₙ∈ℤ exp(πin²τ + 2πinz) is related to elliptic functions and to the Riemann zeta function. At τ = i, ϑ₃(0;i) = Γ(¼)/(√2·π^(3/4)); by Γ(¼)Γ(¾) = π√2 this is exactly the same as π^(1/4)/Γ(¾), not a competing value. Its modular transformation supplies the theta identity used in the zeta functional equation. A function, uncolored.

Cites: [17.20](ch17-refs.md#r-17.20)

See also: [the functional equation](ch18-cards.md#g-funceq), [gamma of one quarter](ch04-cards.md#g-gamma14), [square root of two](ch04-cards.md#g-sqrt2), [Riemann zeta function](ch18-cards.md#g-zeta)

### the classical orthogonal polynomials — `P_{n}(x), H_{n}(x), L_{n}(x)` {#g-orthopoly}

*Function or operator*

Legendre, Hermite, and Laguerre: polynomial families orthogonal under weights, used for spherical harmonics, the quantum oscillator, and the hydrogen atom.

Legendre, Hermite, and Laguerre polynomials are each orthogonal under their own weight. DLMF’s physical applications show Hermite polynomials in harmonic-oscillator eigenfunctions and generalized Laguerre polynomials in hydrogenic radial eigenfunctions; Legendre polynomials are tied to spherical harmonics. Functions, uncolored.

Cites: [17.21](ch17-refs.md#r-17.21)

### the incomplete gamma function — `Γ(s, x)` {#g-incgamma}

*Function or operator*

The gamma integral run from x to infinity instead of from 0, its lower companion γ(s, x) stopping at x; the error function and the chi-squared distribution are both cases of it.

The upper incomplete gamma function is Γ(s, x) = ∫ₓ∞ t^(s−1)e^(−t)dt; its lower companion γ(s, x) runs from 0 to x, and their sum is Γ(s). The identity γ(½, x²) = √π erf(x) connects it to the error function, while regularized incomplete-gamma functions give chi-square distribution functions. A function, uncolored.

Cites: [17.22](ch17-refs.md#r-17.22)

See also: [the error function](ch17-cards.md#g-erffn), [the gamma function](ch17-cards.md#g-gammafn)

### the gamma function — `Γ` {#g-gammafn}

*Function or operator*

Extends the factorial, Γ(n + 1) = n!: Euler’s integral ∫₀^∞ e^(−t) t^(z−1) dt where z has positive real part, continued to the whole plane, never zero, with poles at 0, −1, −2, ….

Euler’s integral, Γ(z) = ∫₀^∞ e^(−t) t^(z−1) dt for Re z > 0, extends by analytic continuation to a meromorphic function on the whole plane with no zeros and a simple pole at each nonpositive integer, the pole at −n having residue (−1)ⁿ/n!. It extends the factorial: Γ(1) = 1 and n! = Γ(n + 1). Euler introduced it in 1729 correspondence with Goldbach, and the symbol Γ is Legendre’s, from his volume of ~1811. Infinitely many smooth functions interpolate the factorial, so why this one? Bohr and Mollerup answered that in 1922: Γ is the only positive function on x > 0 with f(1) = 1 and f(x + 1) = x f(x) whose logarithm is convex. It is the family this chapter’s first brace is drawn from. A function, not a number, so uncolored.

Cites: [17.2](ch17-refs.md#r-17.2), [17.26](ch17-refs.md#r-17.26)

See also: [analytic continuation](ch18-cards.md#g-analcont), [counting notation](ch06-cards.md#g-permfact)

### the Bessel functions — `Jₙ` {#g-besselfn}

*Function or operator*

The solutions of z²w″ + zw′ + (z² − ν²)w = 0: the shapes a circular drumhead can hold.

Bessel’s equation arises when Laplace’s equation or the Helmholtz equation is separated in cylindrical coordinates, so its solutions are the radial shapes of round things that vibrate or conduct heat: a drumhead, a waveguide, a cylinder. Jₙ, the Bessel function of the first kind, is the solution that stays finite at the center, given by an ascending series in which the order ν sets the power of z it starts from. That one fact decides the two values this chapter uses: J₀(0) = 1, because the series for order zero begins with a constant term, while every Jₙ with n ≥ 1 begins with a power of z and so starts from nothing. The zeros of J₀ are the axisymmetric notes a circular drum can play, and the first of them, about 2.4048, is the fundamental. A family of functions, uncolored.

Cites: [17.28](ch17-refs.md#r-17.28), [17.8](ch17-refs.md#r-17.8)

See also: [first zero of the Bessel function J₀](ch17-cards.md#g-j01)

### the Airy functions — `Ai, Bi` {#g-airyfn}

*Function or operator*

The two standard solutions of w″ = zw, the simplest equation whose behavior changes character at a point.

The Airy equation, w″ = zw, is about as simple as a differential equation can be while still being interesting, and what makes it interesting is that the sign of z flips the answer’s character: to the left of the origin the solutions oscillate, to the right one of them decays and the other grows. That is the mathematics of a turning point—a light ray at a caustic, a rainbow’s edge, a quantum particle meeting a linear potential it does not have the energy to climb. Ai and Bi are the two standard solutions, both entire, and Ai is the one that decays. Its value at the origin, 1/(3^(2/3)Γ(2/3)), is one of this chapter’s numbers. A family of functions, uncolored.

Cites: [17.15](ch17-refs.md#r-17.15), [17.28](ch17-refs.md#r-17.28)

See also: [Airy function at zero](ch17-cards.md#g-ai0)

### the error function — `erf` {#g-erffn}

*Function or operator*

erf z = (2/√π) ∫₀^z e^(−t²) dt: the bell curve, integrated, and normalized so that it runs to 1.

The Gaussian e^(−t²) has no antiderivative in elementary terms, so the integral of it is given a name and becomes a function in its own right. The 2/√π in front is chosen so that erf runs from −1 to 1, which makes it the cumulative normal distribution in disguise. Its complement erfc z = 1 − erf z is the tail, the quantity a physicist actually wants when asking how much is left beyond a threshold. The chapter uses erf(1), the probability inside √2 standard deviations. A family of functions, uncolored.

Cites: [17.14](ch17-refs.md#r-17.14), [17.28](ch17-refs.md#r-17.28)

See also: [error function at one](ch17-cards.md#g-erf1), [square root of two](ch04-cards.md#g-sqrt2)

### the Fresnel integrals — `C, S` {#g-fresnelfn}

*Function or operator*

C(z) = ∫₀^z cos(πt²/2) dt and S(z) = ∫₀^z sin(πt²/2) dt: the two coordinates of the spiral light draws at an edge.

Integrate a cosine and a sine whose argument grows as the square of the variable, and plot one against the other: the curve is the Cornu spiral, which winds tighter and tighter toward a point without ever arriving. That point is (½, ½), which is why the chapter’s Fresnel value is a half. The spiral is not a decoration: Cornu drew it in 1874 so that the diffraction of light past a straight edge could be read off a graph. A family of functions, uncolored.

Cites: [17.17](ch17-refs.md#r-17.17), [17.28](ch17-refs.md#r-17.28)

See also: [the Cornu spiral](ch17-cards.md#g-cornu)

### the Cauchy principal value — `p.v. ∫` {#g-pvalue}

*Function or operator*

Cut a symmetric notch out around the bad point and shrink it. Two infinities that would each be undefined cancel, and the integral has a value after all.

An integral whose integrand blows up inside the interval may have no ordinary value: the piece to the left and the piece to the right can each be infinite. The Cauchy principal value removes a symmetric interval around the singularity, integrates what is left, and lets the interval shrink to nothing; because the two halves are cut away at exactly the same rate, their infinities cancel and a finite limit can survive. The symmetry is the whole content—cut asymmetrically and any answer at all can be arranged. It is what makes the logarithmic integral li(x) defined past t = 1, where ln t vanishes, and it is how the distribution p.v.(1/x) is defined. A convention of the framework, uncolored.

Cites: [17.29](ch17-refs.md#r-17.29)

See also: [logarithmic integral](ch17-cards.md#g-li)

### the Lambert W function — `W(x)` {#g-lambertw}

*Function or operator*

The Lambert W function is the multivalued inverse of w ↦ w·e^w, used to solve equations where the unknown appears both inside and outside an exponential.

W(x) answers the question: for which w does w·e^w = x? Because w·e^w is not one-to-one on the reals, W is multivalued; its principal branch W₀ is real for x ≥ -1/e, and a second real branch W₋₁ exists for -1/e ≤ x < 0, meeting W₀ at the branch point (-1/e, -1). The function solves equations from self-referential exponentials, such as x^x = z, to problems in combinatorics and delay differential equations. Lambert studied the related equation in 1758 and Euler solved a special case in 1783; Corless and collaborators standardized the modern notation and theory in 1996.

Cites: [17.30](ch17-refs.md#r-17.30)

See also: [the real numbers](ch01-cards.md#g-reals)

### the beta function — `B(a, b)` {#g-betafn}

*Function or operator*

The beta function B(a, b) = Γ(a)Γ(b)/Γ(a+b) is the two-variable relative of the gamma function, equal to the integral of t^(a−1)(1−t)^(b−1) from 0 to 1.

Euler introduced the beta integral, ∫₀¹ t^(a−1)(1−t)^(b−1) dt, which equals Γ(a)Γ(b)/Γ(a+b), tying it directly to the gamma function. It appears throughout probability, where it normalizes the beta distribution, and in physics, in the Veneziano amplitude that helped launch string theory. The special value B(½, ½) = π, which has a card of its own, follows from Γ(½)² = π.

Cites: [17.31](ch17-refs.md#r-17.31)

See also: [the gamma function](ch17-cards.md#g-gammafn)

### the digamma function — `ψ(x)` {#g-digammafn}

*Function or operator*

The digamma function ψ(x) = Γ′(x)/Γ(x) is the logarithmic derivative of the gamma function.

ψ is defined as Γ′/Γ, and it obeys its own recurrence ψ(x+1) = ψ(x) + 1/x, mirroring the gamma function’s own functional equation. Its special values include ψ(1) = −γ, the negative of the Euler–Mascheroni constant, and ψ(n+1) = Hₙ − γ for positive integers n, tying it to the harmonic numbers. It is the first of the polygamma functions, the higher derivatives of log Γ, and it has an asymptotic expansion of its own, ψ(z) ∼ ln z − 1/(2z) − …, set beside Stirling’s series for the logarithm of the gamma function.

Cites: [17.33](ch17-refs.md#r-17.33)

See also: [the digamma function at one](ch17-cards.md#g-digamma1), [the gamma function](ch17-cards.md#g-gammafn), [harmonic numbers](ch04-cards.md#g-harmonicnum)

### the Cornu spiral {#g-cornu}

*Non-real object*

The Cornu spiral is the curve traced by plotting the Fresnel integrals C(t) against S(t) as t runs from −∞ to ∞, winding into a tight double spiral around the two points (½, ½) and (−½, −½).

Its curvature is proportional to its arc length, a property Leonhard Euler studied in 1744, so it is also called the Euler spiral; Alfred Cornu later used it to analyze diffraction, and the diffraction of light passing a straight edge can be read off its graph. The two points the curve winds toward, at (½, ½) and (−½, −½), are the limiting values of the Fresnel integrals themselves. Outside optics the same curve, sometimes called a clothoid, is used to design smooth, jerk-free transitions in roads and railway track.

Cites: [17.35](ch17-refs.md#r-17.35)

See also: [the Fresnel integrals](ch17-cards.md#g-fresnelfn)