Asymptotic series & transseries — flashcards

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.

Fractional integration by parts, derived and verified using the Grünwald–Letnikov finite-difference definition of the fractional derivative (α=0.6: both sides of the identity matched to 10 significant digits, 0.22554001162413295 vs. 0.22554001162413304), establishes a genuine asymmetry absent at integer order: a left-sided fractional derivative under integration by parts produces a right-sided one. The resulting fractional Euler–Lagrange equation (Agrawal, 2002) mixes left and right fractional derivatives for exactly this reason — Riewe's original motivation (1996–97) was building genuine dissipation directly into a Lagrangian via this left/right mixing, rather than bolting on an external friction force.

A Legendre transform applied to a fractional Lagrangian carries the left/right derivative asymmetry from st/fractional-calculus-of-variations into fractional Hamilton's equations: position evolves by a left ('backward-looking') fractional derivative, momentum by a right ('forward-looking') one. The Mittag-Leffler function E_α(z)=∑z^k/Γ(αk+1), the natural fractional generalization of e^z, was verified as an exact eigenfunction of the Caputo fractional derivative (D^α_C E_α(λt^α)=λE_α(λt^α)) by differentiating the defining series term by term, matching to 20 digits. This machinery was applied directly to the Berry–Keating conjecture — H=xp as a candidate classical Hamiltonian for the Hilbert–Pólya program, whose ordinary classical dynamics x(t)=x_0e^t, p(t)=p_0e^{-t} was independently verified (x(3)=20.0855 against predicted x_0e^3=20.0855, H conserved to floating-point precision) — giving fractional Hamilton's-equation solutions in terms of E_α rather than ordinary exponentials, with long-time behavior turning power-law rather than exponential for 0<α<1.

Laskin's fractional Schrödinger equation — motivated by path integrals over Lévy flights rather than Brownian motion, replacing the ordinary Laplacian kinetic term with a fractional one |k|^α defined via Fourier multiplication — reuses the identical construction (replace an ordinary derivative's Fourier multiplier with |k|^α) already established for the fractional-derivative machinery in st/fractional-calculus-of-variations and st/fractional-hamiltonian-and-berry-keating, applied here to a new, quantum-mechanical setting rather than as a new mathematical tool. Time evolution under this equation was tested for linearity — the one structural property the no-cloning theorem actually depends on — via a split-step Fourier simulation: U(aψ₁+bψ₂) vs. aU(ψ₁)+bU(ψ₂) matched to 1.2×10⁻¹⁵ under α=1.4 evolution.

Ramanujan's Master Theorem: if f(x)=∑(−1)^k/k!·φ(k)·x^k, then the Mellin transform M[f](s)=Γ(s)φ(−s) — verified on f(x)=e^{−x} (φ≡1, recovering Γ(s) itself) and f(x)=1/(1+x) (φ(k)=k!, recovering Γ(s)Γ(1−s)), both matching to high precision. Carlson's theorem (a function of sufficiently controlled exponential growth that vanishes at every nonnegative integer must vanish identically) is the rigorous justification behind Ramanujan's own formal derivation. Why Γ enters by multiplication rather than division — unlike ordinary Taylor's theorem, which divides by k! to compensate for repeated differentiation — resolves once you notice the 1/k! is already part of the given data, not something the theorem introduces: a Mellin–Barnes contour shift shows Γ(−s)'s poles at every nonnegative integer k have residue exactly (−1)^(k+1)/k! (verified for k=0,…,4), so the entire infinite family of reciprocal factorials is already encoded in Γ's own residue structure, and residue-sum reconstruction of the original series is what recovers the 1/k! normalization — not a division anywhere.

For CUE, the maximum of the log-characteristic polynomial obeys max_θ log|P_N(θ)| = log N − (3/4)log log N + O(1) — proved (Arguin–Belius–Harper) — with the −(3/4)log log N correction term the precise signature distinguishing a log-correlated field's extremes from an independent (REM-type) field's, and the whole result an instance of branching Brownian motion universality. The analogous statement for max|ζ(1/2+it)| over a bounded range is conjectured, not proved, by the same mechanism.

The freezing transition (st/freezing-transition-log-correlated-extremes) produces Gumbel-distributed extreme-value statistics; the Gumbel distribution's skewness constant, 12√6·ζ(3)/π³, was derived here from first principles rather than looked up, by identifying the Gumbel distribution's moment generating function as literally Γ(1−t) — so its cumulant generating function is the same log-Gamma Taylor series whose coefficients are zeta values at integers. This ties Apéry's constant ζ(3)'s appearance in the freezing transition's maximum-value statistics directly to the digamma/zeta coefficient identity already established for the completed zeta function's log-derivative (st/xi-log-derivative-local-global-split).

Analytically continuing the Keating–Snaith formula to complex moment-order β reveals a genuine pole at β=−1/2 — an exact divergence of the formula itself, not merely an asymptotic artifact — traced to the elementary local level-repulsion exponent governing near-coincidence of eigenvalues; confirmed via the argument principle, a small contour around β=−1/2 in the complex moment-order plane having winding number exactly −1. Purely imaginary β=iτ computes something structurally different: the characteristic function of log|det(I−U)|, verified by direct Monte Carlo, with the complete absence of phase-transition behavior on that axis (boundedness by 1 at every τ tested, smooth decay with N) following directly from the fact that a characteristic function of a real random variable cannot diverge.

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.

Émile Borel, 1899.

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.

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.

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.

Expansions that diverge for every argument and are nonetheless the best finite description available. Coefficients grow factorially rather than geometrically, which is why the divergence has a different cause from a finite radius of convergence, and why Borel summation and resurgence exist.