Foundations — Full Fourier series — coefficients derivation

How to read this page

The parent note uses a lot of notation as if you already knew it: $f(x)$, $[-L,L]$, $\sin \frac{n\pi x}{L}$, ${\int }_{-L}^L$, "odd/even", "orthogonality", $a_0/2$. Below we earn every one of them from zero, each building on the one before. A smart 12-year-old should be able to start at line one.

1. $x$ and $f(x)$ — the input and the output

The picture: a curve on a graph. The horizontal axis is $x$; the height of the curve above each $x$ is $f(x)$.

Why the topic needs it: the thing we want to rebuild — a sound wave, a temperature profile, a voltage — is exactly such a curve $f(x)$.

2. Period and the interval $[-L,L]$

The symbol $[-L,L]$ just means "all the $x$ values from $-L$ up to $L$, endpoints included." The letter $L$ is a half-period: half the width of one copy. If the period is $2\pi$, then $L=\pi$ (that is Example 2 in the parent).

The picture: one tile of a wallpaper pattern, from $-L$ on the left to $+L$ on the right; the whole wall is that tile stamped over and over.

Why the topic needs it: we only ever study one period. If we get $f$ right on $[-L,L]$, periodicity fills in everything else for free.

3. Sine and cosine as spinning height and width

As $\theta$ grows, both trace smooth up-and-down waves. They are the purest possible wiggles — one hump up, one hump down, repeat.

4. Frequency: the meaning of $\frac{n\pi x}{L}$

We do not just use $\sin x$; we use $\sin \frac{n\pi x}{L}$. Let us unpack that angle piece by piece.

• $\frac{\pi x}{L}$: as $x$ runs from $-L$ to $L$, this angle runs from $-\pi$ to $\pi$ — exactly one full wave across one period. This is the slowest wave, $n=1$.
• The integer $n=1,2,3,\dots$ is the harmonic number. Multiplying the angle by $n$ makes the wave wiggle $n$ times faster, so $n$ complete waves fit in one period.

Why the topic needs it: every one of these has period dividing $2L$, so any sum of them repeats every $2L$ — matching $f$. Different $n$ = different pitch; the coefficients say how loud each pitch is.

5. The summation sign $\sum _{n=1}^{\infty }$

The picture: a stack of waves laid on top of each other, their heights added at every $x$ to make the final curve.

Why the topic needs it: the Fourier series $f(x)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }[a_n\cos \frac{n\pi x}{L}+b_n\sin \frac{n\pi x}{L}]$ is such an endless sum. The $a_n,b_n$ are the loudness knobs.

6. The coefficients $a_0,a_n,b_n$

Think of a graphic-equaliser: each $a_n$ and $b_n$ is one slider setting.

7. The integral ${\int }_{-L}^{L}g(x)dx$ — "signed area"

The picture: shade the region under the curve. Above the axis = "+" shading, below = "−" shading. The integral is (+ shading) minus (− shading).

8. Even and odd functions — the symmetry shortcut

Two facts do heavy lifting:

• An odd function has equal + and − area over $[-L,L]$, so ${\int }_{-L}^L(\text{odd})dx=0$.
• $\text{even}\times \text{even}=\text{even}$, $\text{odd}\times \text{odd}=\text{even}$, $\text{even}\times \text{odd}=\text{odd}$ (same rule as sign multiplication).

Why the topic needs it: if $f$ is odd, every $a_n$ vanishes before you compute anything (Example 1 and 2 in the parent). If $f$ is even, every $b_n$ vanishes. This is why the parent says "check symmetry first." See Even and odd functions.

9. Orthogonality — waves that don't overlap

10. The boxed formulas we are heading toward

Prerequisite map

Related vault topics

Once these foundations are solid, the same machinery powers Half-range Fourier series (sine and cosine), Complex (exponential) Fourier series, Parseval's theorem, Convergence of Fourier series (Dirichlet conditions), and Separation of variables for the heat equation.

Equipment checklist

Cover the right-hand side and test yourself. Each ::: line is a self-check.