# Workbook 9 — Companion: the even reduction > [!abstract] What this page is for > Workbook 9 hands you the sawtooth's complex coefficients and asks for the algebraic reduction to real sinusoids: split the doubly infinite sum, reindex, and let a conjugate-pair identity collapse each pairing. This page runs **the same three moves on a different function** — one whose coefficients are *even* in $n$, so the *other* conjugate-pair identity fires and cosines come out instead of sines. Follow it end to end and the sheet's sines are still yours to extract. > > 📄 [[MATH310F26_WB9.pdf|Workbook 9 (blank, 1 page)]] · [[MATH310F26(Day 10) - Hearing the Fourier transform|Day 10 note]] · [[MATH310F26(Workbook 6 Companion) - What a conjugate pair extracts|WB6 companion]] > > 🎓 [Submit on Canvas](https://elearning.mines.edu/courses/81632/assignments/587523) --- ## 1. The given: a tent and its coefficients Take $f(x)=|x|$ on $(-\pi,\pi)$, extended $2\pi$-periodically — a tent tiled into a **triangle wave**, with corners at the seams but *no jumps*: the one-sided values at $x=\pm\pi$ agree. A Monday-style computation (orthogonality plus one integration by parts — worth redoing when the sheet is done) delivers the complex coefficients: $c_{0}=\pi^{2},\qquad c_{n}=\frac{2\big((-1)^{n}-1\big)}{n^{2}}\ \ (n\neq 0) \qquad\text{so}\qquad c_{n}=\begin{cases}-\,4/n^{2}, & n\ \text{odd},\\ 0, & n\ \text{even},\ n\neq 0.\end{cases}$ Two structural facts to bank before reducing. The coefficients are **real**, and they are **even in $n$**: $c_{-n}=c_{n}$, because $(-1)^{-n}=(-1)^{n}$ and $(-n)^{2}=n^{2}$. Compare the sheet: the sawtooth's $c_{n}=\tfrac{2\pi(-1)^{n}}{n}i$ are *imaginary* and *odd* in $n$. That sign structure is the whole story of what comes out. ## 2. The three moves **Split and reindex.** The representation $f(x)=\sum_{n=-\infty}^{\infty}\frac{c_{n}}{2\pi}e^{inx}$ keeps its $n=0$ term this time — $\frac{c_{0}}{2\pi}=\frac{\pi^{2}}{2\pi}=\frac{\pi}{2}$, a genuine DC offset, sensible for a function that is never negative. Split the rest into $n\geq 1$ and $n\leq -1$ and reindex the second half by $n\mapsto -n$. Because $c_{-n}=c_{n}$, the two halves share one coefficient: $f(x)=\frac{\pi}{2}+\sum_{n=1}^{\infty}\frac{c_{n}}{2\pi}\Big(e^{inx}+e^{-inx}\Big).$ **Collapse the pair.** The sheet's pairing produced a *difference* of exponentials and called on $e^{i\theta}-e^{-i\theta}=2i\sin\theta$. Even coefficients produce a **sum**, and the partner identity fires: $e^{inx}+e^{-inx}=2\cos(nx)$ — no stray $i$ to chase, because nothing here was imaginary. **Simplify.** With $\frac{c_{n}}{2\pi}=\frac{(-1)^{n}-1}{\pi n^{2}}$, the even terms vanish and the odd ones carry $-2$: $f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n\ \mathrm{odd}}\frac{\cos(nx)}{n^{2}}.$ ![[wb9_guide_triangle.png]] ## 3. Read the result against the sheet's Everything mirrors. The sawtooth is **odd**, its coefficients imaginary-and-odd, and the reduction returned **sines only** with no constant. The triangle is **even**, its coefficients real-and-even, and the reduction returns **cosines only** plus a DC term. And the equals sign behaves differently too: the sawtooth *jumps* at the seams, its coefficients fade like $1/n$, and its truncations ripple; the triangle is continuous, its coefficients fade like $1/n^{2}$, and by $S_{9}$ the sum is visually indistinguishable from the tent — evaluate at the seam $x=\pi$ and the series honestly returns $\pi$, no averaging clause required. Smoothness in $x$ *is* decay speed in $n$; your ears met this on Friday as triangle-mellow versus sawtooth-icky. ## 4. What to take to the sheet The same three moves, with the signs flipped: your coefficients are odd in $n$, so the reindexed half picks up a minus, the pairing is a *difference*, and the $2i\sin$ identity plus one $i\cdot 2i$ does the collapse. When your series lands, run the sheet's last item knowing what to expect at a *jump* — this page's function never had one, and that contrast is the lesson. > [!success]- Numbers checked in one sandbox run > The given coefficients: $\int_{-\pi}^{\pi}|x|e^{-inx}dx$ returned $9.869604=\pi^{2}$, $-4.000000$, $0.000000$, $-0.444444$, $0.000000$, $-0.160000$ for $n=0,\dots,5$ — matching $2((-1)^{n}-1)/n^{2}$ exactly. The reduced series evaluated to $0.3,\ 1.9,\ 2.5$ at $x=0.3,\ 1.9,\ -2.5$ (2000 terms), and at the seam $x=\pi$ returned $3.141274\to\pi$ — convergence *at* the corner, jump-free.