Differentiation & integration — flashcards

The finite-difference argument that produces a polynomial for positive powers fails for negative ones, because the falling factorials no longer span. The discrete and continuous cases fail for different reasons, which is why the harmonic numbers have an asymptotic expansion but no closed form.

An entire function of controlled growth is determined by its zeros up to an exponential factor. This — completeness of the zero set plus a growth bound — is what licenses Euler's product for sine, not the fact that the zeros happen to be evenly spaced.

ζ(2) = π²/6, obtained by reading the second coefficient of the sine product as an elementary symmetric function of the reciprocal zeros. The inference runs from coefficients to statistics of the roots, not the other way round.

A Taylor series has a finite radius of convergence because its coefficients grow geometrically, controlled by the nearest singularity. An asymptotic series diverges everywhere because its coefficients grow factorially. Different mechanisms; Borel summation is what recovers a function from the second.

Effects mattering at every scale is not sufficient to make fractal machinery the right tool; what is required is invariance under a scaling with a power law. Multiscale relevance without self-similarity is the domain of renormalisation, not of dimension theory.

Fractional Noether's theorem (Frederico–Torres, 2007) states that the conserved quantity satisfies a fractional differential condition rather than ordinary constancy — whether this still forces genuine constancy is convention-dependent: the Caputo derivative's kernel is exactly the constants, while the Riemann–Liouville derivative's kernel is larger, including non-constant power-law functions. The inverse-square potential V(x)=k/x² was identified and verified as the unique power-law potential whose full action (not just its equations of motion) has an exact dilation symmetry x→λx, t→λ²t; the resulting conserved quantity D=tH−½xp was verified numerically conserved to ~2×10⁻⁸ along an integrated trajectory, alongside H itself. The connection to conformal quantum mechanics (de Alfaro–Fubini–Furlan, 1976) and the full SO(2,1) symmetry (energy, dilation, and a third 'special conformal' generator) was named, with only D itself independently verified — the third generator and the full algebra were not constructed or checked here.

The fine structure constant α is a running coupling, not a scale-invariant quantity: verified numerically at two energy scales, α⁻¹(0)=137.036 against α⁻¹(M_Z)=128.957 (measured values), with a simple one-electron-loop estimate reproducing α⁻¹(M_Z)≈134.47 — capturing part but not all of the shift, the remainder named (muon, tau, hadronic vacuum polarization) rather than computed. This was deliberately set up as the inverted case relative to st/fractional-noether-and-dilation-current's 1/x² potential: both ask the identical diagnostic question — does this system have an exact dilation symmetry, evidenced by a genuine conserved current, or does the relevant coupling run with scale instead — with the 1/x² potential answering 'exact symmetry' and the fine structure constant answering 'runs' via the QED beta function (vacuum polarization screening).

A series converges in ℚ_p if and only if its terms tend to zero. The condition that is merely necessary over ℝ is sufficient here, and the harmonic series has no p-adic analogue of its failure.

exp_p converges for v_p(x) > 1/(p−1) and log_p(1+x) for v_p(x) ≥ 1; on their common domain they invert one another and give (1 + pℤ_p, ×) ≅ (pℤ_p, +) as topological groups. p = 2 needs the smaller disc.

Log-differentiating Euler's sine product sin(πx) = πx·∏(1−x²/n²) once gives the cotangent partial-fraction expansion π cot(πx) = ∑ 1/(x−n); differentiating again gives π²/sin²(πx) = ∑ 1/(x−n)² — exactly the lighthouse formula (st/lighthouse-reciprocal-square). Wallis' product is instead the x=1/2 evaluation of the original, undifferentiated product. So the lighthouse/Basel identity and Wallis' product are two different operations — differentiate twice, versus evaluate at 1/2 — performed on the same underlying product, not two independent facts that happen to coincide.

The Wallis product 2/1·2/3·4/3·4/5·⋯ = π/2 has at least three independent derivations that manage error differently, not three restatements of one argument. (1) Analytic: it is the x=1/2 evaluation of Euler's undifferentiated sine product sin(πx) = πx·∏(1−x²/n²) (st/basel-wallis-same-product) — a substitution into an already-established product, with no limiting process of its own beyond the product's own convergence. (2) Geometric/algebraic: N evenly-spaced points on a circle as the roots of x^N−1, giving two exact finite-N lemmas (observer halfway between two points: distance product = 2; observer replacing one point: distance product = N) whose ratio N·(distance)/2 is an *exact* invariant at every finite N — a single limit, taken once at the very end as N→∞, converts that invariant into π/2 via chord-length-to-arc-length convergence. (3) Classical calculus: Wallis's own route via the reduction formula ∫₀^(π/2) sin^n θ dθ, giving the asymptotic (2n)!!/(2n−1)!! ∼ √(πn) — which stacks two separate approximations (a pointwise limit inside the reduction, and an asymptotic equivalence) that must cancel jointly, rather than carrying one exact quantity to a single final limit. Routes (2) and (3) both pass through the same trigonometric object — a quarter-circle arc — entered through different doors: chord geometry directly, versus a sine substitution turning an algebraic integral into an arc-length integral.

st/wallis-product-three-routes' route (2) sums two individually divergent halves (terms from either side of the observer) into a finite limit. A natural conjecture, by direct analogy to the Riemann rearrangement theorem, is that any interleaving drawing from both halves converges to the same value, since order shouldn't matter as long as both sides are eventually included. This is false in a precise, quantified way: a balanced one-for-one interleaving gives π/2 (at f=1/2), but a 2-negative-to-1-positive interleaving gives a different, still-finite limit, (π/2)·2^(−1/2) — and the general p:1 interleaving carries an exact correction factor of p^(−f), derived from the two halves' asymptotic growth rates and confirmed numerically to six digits. Unlike the Riemann rearrangement theorem, where order can send a conditionally convergent series to any real value (or make it diverge) with no simple rule connecting the rearrangement to the outcome, this construction stays finite under every fixed-ratio interleaving and obeys one clean, computable law for exactly how the order matters.

The Korteweg–de Vries equation is the motivating case for a 'nonlinear Fourier transform': inverse scattering (Gardner–Greene–Kruskal–Miura, 1967) linearizes KdV via an auxiliary linear scattering problem, in a role structurally analogous to the ordinary Fourier transform for linear PDEs but through a genuinely different mechanism — a Lax pair (L,A) with L̇=[A,L] (Lax, 1968). The general mechanism is elementary once stated: L̇=[A,L] implies L(t)=U(t)L(0)U⁻¹(t) for a unitary U(t) built from A, so eigenvalues are conserved by pure linear algebra — for any A(t) whatsoever, nothing specific to KdV. This is the same algebraic fact as the Heisenberg equation of motion, Ȯ=(i/ℏ)[H,O]. The single-soliton solution u=2κ²sech²(κ(x−4κ²t)) was verified to satisfy KdV exactly (symbolic residual reduces to 0), and its associated bound-state eigenfunction ψ=sech(κx) verified against −ψ''+V(x)ψ=−κ²ψ.

Leonhard Euler, 1735.

Karl Weierstrass, then Jacques Hadamard, 1876 (Weierstrass), 1893 (Hadamard).

Leonhard Euler, 1735.

Émile Borel, 1899.

Rufus Bowen, 1979.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

An asymptotic series remembers a function to every finite order and forgets it at all orders at once. The content is in how fast the coefficients grow — factorially, not geometrically — so the divergence is the message rather than the failure.

The reals are the completion at the infinite place. Being the ambient number system you happened to grow up with is a fact about pedagogy, not about ℝ.

A convergent power series is the transseries in which every exponentially small term happens to vanish. Convergence is a degenerate case of expansion, not the definition of it.

Ordinary analysis is what you get by taking the one branch we happen to live on. Naming it a branch rather than the default is the whole point of the lens.

REFUTED. The two divergences look alike but come from different mechanisms: a Taylor series' radius of convergence is set by geometric growth of coefficients, controlled by the nearest singularity; an asymptotic series diverges everywhere because its coefficients grow factorially. Different mechanism, not the same phenomenon wearing two names.

Recorded as refuted: this edge was drawn and then withdrawn.

`Mathlib.NumberTheory.Ostrowski` imports `Mathlib.Analysis.AbsoluteValue.Equivalence`, `Mathlib.Analysis.SpecialFunctions.Log.Base`, `Mathlib.Analysis.SpecialFunctions.Pow.Continuity`. Read off Mathlib/NumberTheory/Ostrowski.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.Hensel` imports `Mathlib.Analysis.SpecificLimits.Basic`, `Mathlib.Analysis.Normed.Field.ProperSpace`. Read off Mathlib/NumberTheory/Padics/Hensel.html. No human asserted this edge.

The calculus of elementary functions — the core toolkit before measure theory or real analysis proper.