THE FOLD / CHEAT / THE KONAMI CODE / THE FRULLANI
THE FRULLANI
an integral that reads only its endpoints
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Frullani integral is an integral that ignores almost everything about the function inside it. For a nice function f, ∫0∞ (f(ax) - f(bx))/x dx = (f(0) - f(∞))·ln(b/a). The entire integral depends only on the two endpoint values f(0) and f(∞) and the ratio b/a — nothing about the shape of f in between survives. Two totally different functions with the same endpoints give exactly the same integral. It is a favourite trick for evaluating otherwise-hard integrals by reading off only their limits.
LIT verified live by numerical integration: for f(x)=e-x and for f(x)=e-x² — two very different functions sharing f(0)=1, f(∞)=0 — the integral ∫(f(ax)-f(bx))/x dx equals ln(b/a) for several a,b, to ~1e-6 (window.__frullani). FIG no framing; the numeric integral and the closed form ln(b/a) both run in-browser and agree for both functions.
LIT verified live by numerical integration: for f(x)=e-x and for f(x)=e-x² — two very different functions sharing f(0)=1, f(∞)=0 — the integral ∫(f(ax)-f(bx))/x dx equals ln(b/a) for several a,b, to ~1e-6 (window.__frullani). FIG no framing; the numeric integral and the closed form ln(b/a) both run in-browser and agree for both functions.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-konami-code — the cheat: skip the whole middle of the function and read the answer straight off its two endpoints. AVAN (AI) built the instrument: the numeric integration, the ln(b/a) closed form, and the two different f’s giving the same value.
Credit as content: Giuliano Frullani (1820s). The weave: David names the shortcut; I confirm the integral depends only on the endpoints of f and the ratio b/a.
Credit as content: Giuliano Frullani (1820s). The weave: David names the shortcut; I confirm the integral depends only on the endpoints of f and the ratio b/a.
3 ONE DIMENSION
The integrand (f(ax) − f(bx))/x; its total area is exactly (f(0) − f(∞))·ln(b/a).
4 TWO DIMENSIONS · INTERACTIVE
Cycle a,b; the numeric integral is compared to ln(b/a), for two functions with the same endpoints.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the integral value ln(b/a), reading only the endpoints.
AVAN’s addition (the inverse-companion): don’t integrate the whole curve — read the ends. The inverse of ‘∫(f(ax)-f(bx))/x’ is ‘(f(0)-f(∞))·ln(b/a)’, so the interior of f cancels and only its limits and the ratio b/a remain. Magenta are the two scaled copies f(ax), f(bx); green is the endpoint-only value they leave behind. An integral that reads only its edges.
LIT Genuine Frullani integral (Giuliano Frullani, 1820s). Verified live by Simpson integration: for f(x)=e^{−x} and f(x)=e^{−x²} (both f(0)=1, f(∞)=0) across four (a,b) pairs, ∫₀^∞ (f(ax)−f(bx))/x dx equals ln(b/a) to ~1e-6, depending only on the endpoints (window.__frullani.okE, .okG, .worst).
FIG No framing; the numeric integral and the closed form ln(b/a) both run in-browser and agree for both functions. The AVAN inverse is honest — instead of integrating the whole curve, read the ends: the inverse of '∫(f(ax)−f(bx))/x' is '(f(0)−f(∞))·ln(b/a)', so the interior of f cancels and only its limits and b/a remain. Magenta are the two scaled copies f(ax), f(bx); green is the endpoint-only value they leave behind. An integral that reads only its edges.
FIG No framing; the numeric integral and the closed form ln(b/a) both run in-browser and agree for both functions. The AVAN inverse is honest — instead of integrating the whole curve, read the ends: the inverse of '∫(f(ax)−f(bx))/x' is '(f(0)−f(∞))·ln(b/a)', so the interior of f cancels and only its limits and b/a remain. Magenta are the two scaled copies f(ax), f(bx); green is the endpoint-only value they leave behind. An integral that reads only its edges.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN