Nothing in this chapter is new. That is deliberately the entire point. Every idea below already has a home somewhere in Calculus II or Linear Algebra — periodic functions, convergence of sequences and series, improper integrals, inner products, orthogonal projection, eigenvectors — and this chapter's only job is to make sure those ideas are loaded and ready before the course leans on them, chapter after chapter, without ever re-deriving them from scratch. A course this cumulative can't afford a shaky foundation three chapters in; better to find the gap here, on review material, than in the middle of deriving a coefficient formula that assumes it.
One piece is genuinely built here rather than merely reviewed: complex numbers and Euler's formula, e^{i\theta}=\cos\theta+i\sin\theta. A future Complex Analysis course exists on this catalog, but it hasn't been written yet, and this course needs complex exponentials starting in chapter 3 regardless. So this chapter derives everything required — converting a real sinusoid to complex-exponential form and back, multiplying complex numbers as a rotation-and-scaling, the handful of algebraic identities the rest of the course reaches for by reflex — as self-contained, ordinary Calculus II and College Algebra material. No later chapter assumes a Complex Analysis course was ever taken.
The rest is a checklist with teeth. Verifying that \sin(mx) and \cos(nx) are orthogonal on an interval by direct integration previews exactly the computation chapter 3's coefficient formulas are built from. Revisiting orthogonal projection in \mathbb{R}^n sets up the single biggest idea this course borrows from Linear Algebra: a Fourier coefficient is nothing but a projection coordinate, in a vector space where the "vectors" happen to be functions. Eigenvalues and diagonalization make one more appearance late in this course, in a setting few readers expect the first time through. Every section here answers one question — is this piece of the toolbox actually sharp — before the chapter that needs it gets to assume the answer is yes.