Advanced Analysis: Feynman + Abel

The Week 8 pattern was: embed, differentiate once, integrate back from an anchor. The advanced repertoire adds three gears. SECOND derivatives, when one differentiation simplifies but does not close — $I''(t)$ sometimes satisfies a differential equation with $I$ itself (the classic $I(t) = \int_0^\infty e^{-x^2}\cos(tx)\,dx$ gives $I'(t) = -\frac{t}{2}I(t)$, an ODE whose solution $I(t) = \frac{\sqrt\pi}{2}e^{-t^2/4}$ evaluates a whole family). Learn to see 'my derivative re-expresses in terms of me' as a WIN, not a failure to simplify.

PARAMETER PLACEMENT is the real skill: put $t$ where differentiation kills the ugliest factor. A $\ln x$ in the integrand wants $x^t$ (since $\partial_t x^t = x^t \ln x$ — the log APPEARS under differentiation, so start one level below it); an $\arctan$ wants its argument parameterized; a stubborn $\frac{1}{x^2 + a^2}$ invites differentiating in $a$ to generate higher powers. You are choosing the family so that the $t$-derivative is the member with the closed form.

The third gear is domain symmetry composed with parameters: substitutions like $x \mapsto \frac{\pi}{2} - x$ or $x \mapsto \frac{1}{x}$ (Week 6's reflection trick) often turn $I(t)$ into $I(\text{something}(t))$, giving a FUNCTIONAL EQUATION for $I$ that replaces integration entirely. The famous $\int_0^{\pi/2} \ln(\sin x)\,dx = -\frac{\pi}{2}\ln 2$ falls to exactly this: reflection plus the double-angle identity yields an equation in the integral itself.

Pitfall. Iterating $2I(t) = I(t^2)$ without the continuity step. The telescoped identity $I(t) = 2^{-n} I(t^{2^n})$ needs $I(t^{2^n}) \to I(0)$, which is continuity at 0 plus boundedness — one sentence, but the proof is incomplete without it (and for $|t| > 1$ the answer is different: $2\pi \ln|t|$, via pulling out $t^2$).

Week 9 used Abel for convergence verdicts; the advanced use is ESTIMATION. Summation by parts converts $\sum_{n \le N} a_n f(n)$ into $A(N)f(N) - \int_1^N A(t) f'(t)\,dt$ (the integral form, with $A(t) = \sum_{n \le t} a_n$) — so any estimate on the partial sums $A(t)$ converts into an estimate on weighted sums against SMOOTH weights. This is how 'the average of $a_n$ is $c$' upgrades to asymptotics for $\sum \frac{a_n}{n}$, $\sum a_n \ln n$, and every similar cousin.

The companion is the integral comparison you already trust, sharpened: for monotone $f$, $\sum_{n=1}^N f(n) = \int_1^N f + O(|f(1)| + |f(N)|)$, and one more order of precision comes from the trapezoid/Euler-Maclaurin correction $\frac{f(1) + f(N)}{2}$. Harmonic numbers ($H_N = \ln N + \gamma + O(1/N)$) and Stirling ($\ln n! = n\ln n - n + \frac{1}{2}\ln(2\pi n) + O(1/n)$) are the two outputs used so often they should be reflexes with their error terms attached.

A5/B5-grade problems love hiding an Abel summation inside a limit: 'evaluate $\lim \frac{1}{N}\sum_{n \le N} a_n b_n$' where $a_n$ oscillates and $b_n$ drifts slowly. The template: partial-sum bound on the oscillator, summation by parts to move the drift's derivative into an integral, and the limit drops out of the leading term. When you see mixed oscillation-and-drift, Abel is not one option — it is the designed route.

Pitfall. Writing $\sum f(n) \approx \int f$ without tracking WHICH endpoint convention and error size you are using. The exam's asymptotics problems hide their difficulty in the error term; the main term is usually free.

A3/B3 problems extend the A2 anatomy by one act: transform, transform AGAIN, then finish — or one transformation plus a genuinely creative construction. First exposure this week comes with modified rules: untimed first attempts, 45-minute soft cap, and the goal is ARC RECOGNITION, not solve rate. A 30% solve rate on first-contact A3s at this stage is on-curve for the December target; what must be 100% is producing a real attack, because partial credit at this level pays 2-4 points and the difference between 0 and 3 across four sessions is a placement band.

The new behavior at this level: problems stop confirming your progress. On A1, a correct move visibly simplifies things; on A3, the correct first move often makes the page LOOK worse (a substitution that doubles the symbols, a generalization that seems gratuitous) and pays off two moves later. The discipline is evaluating moves by structural promise — did it expose symmetry, create an invariant, make an induction possible? — rather than by visual simplicity.

Study protocol for the misses (most first attempts will miss): before reading the solution, write your best partial idea down. Then read ONLY until the solution's first real idea, stop, and try to finish alone. Full-solution reading is a last resort — the interrupted-read protocol trains the skill you actually need, which is CONTINUING from a foothold, and it is how the deep-study gates this phase are designed to be used.

Pitfall. Abandoning a correct move because the page got messier. At A3 depth, evaluate moves by structure exposed (symmetry, telescoping shape, known sum) — visual simplicity returns one act later than you want it to.