Fourier Series — Physica

Equivalent forms

f(x) = \sum_{n=-\infty}^{\infty} c_n\, e^{i n\pi x/L}

c_n = \frac{1}{2L}\int_{-L}^{L} f(x)\, e^{-i n\pi x/L}\, dx

a_n = \frac{1}{L}\int_{-L}^{L} f(x)\cos\frac{n\pi x}{L}\,dx

Orthogonality of sines and cosines turns a function into a discrete spectrum — and the same trick diagonalizes the heat, wave, and Schrödinger equations.

Symbol	Name	SI unit	Dimension	Range
$f$	Partial sumoutput Truncated Fourier reconstruction at point x	1	1	-2 – 2
$N$	Number of harmonics Terms kept in the partial sum	1	1	1 – 50
$x$	Position Coordinate within one period	rad	1	-3.1416 – 3.1416
$L$	Half-period Half the period of f	rad	1	0.5 – 6.283

Unit systems

Symbol	SI	CGS	Imperial
$x$	rad	rad	rad
$L$	rad	rad	rad
$N$	1	1	1
$f$	1	1	1

Where it holds

Periodic f satisfying the Dirichlet conditions (piecewise continuous, finite number of extrema per period).

Dimensional analysis

f(x) \to [f]

\Sigma a_n \cos (n\pi x/L) \to [

a_n]

Coefficients a_n, b_n inherit the units of f; the cosine/sine arguments

n\pi x/L

are dimensionless.

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Discovery

Joseph Fourier · 1822

Studying heat flow, Fourier claimed in 1807 that any function could be written as a sum of sines — a claim so bold that Lagrange resisted it. His 1822 Théorie analytique de la chaleur vindicated the idea and launched harmonic analysis.

Try this

A jagged square wave looks nothing like a smooth sine. Yet stack enough sines and the corners appear out of nowhere — every periodic shape is a chord of pure tones.

Add harmonics one by one and watch a sum of sines snap into a square wave, Gibbs overshoot and all.

Research status: stable

Real-world applications

Audio synthesis and timbre analysis
Solving PDEs by separation of variables
Signal compression (JPEG's cousin, the DCT)
Vibration and harmonic-distortion analysis in engineering

Common misconceptions

Fourier series need the function to be smooth — they handle discontinuous square and sawtooth waves (with Gibbs overshoot).
More terms always reduce error everywhere — near a jump the overshoot height stays $\sim 9$ % no matter how many terms.
The series equals f at the jump — it converges to the midpoint there.

Experimental verification

Every spectrum analyzer, musical-timbre measurement, and harmonic-distortion test confirms the decomposition; the heat-equation solutions Fourier derived match measured temperature relaxation.

Derivation

Sines and cosines are orthogonal over a period:

\int _{-L}^{L} \sin (n\pi x/L)\sin (m\pi x/L)dx = L\delta _{nm}

.

Projecting f onto each basis function — multiply by the basis function and integrate — isolates its coefficient, giving a_

n = (1/L)\int f \cos (n\pi x/L)dx

and similarly for b_n.

Limiting cases

N \to \infty

⟶ Exact reconstructionThe series converges to f at every point of continuity.

At a jump discontinuity⟶ Converges to the midpointThe series gives the average of left and right limits, with a persistent

\sim 9

% Gibbs overshoot nearby.

f smooth and periodic⟶ Coefficients decay fastThe smoother f is, the faster a_n, b_n shrink (spectral accuracy).

What if…

What if the period goes to infinity?

The discrete spectrum becomes continuous and the sum becomes an integral — the Fourier transform.

What if f is even or odd?

Even f keeps only cosines; odd f keeps only sines — symmetry halves the work.

1

Square-wave coefficients

Given ·

f:: odd square wave, $\pm 1$

Find · b_n

Steps

Odd function → only sine terms
$b_n = (2/L)\int _0^L 1\cdot \sin (n\pi x/L)dx$
$= 4/(n\pi )$ for odd n: $f = (4/\pi )(\sin x + \sin 3x/3 + \sin 5x/5$ + …)

Result ·

b_n = 4/(n\pi )

for odd n, 0 for even n

2

Value at x = π/2 (N = 3)

Given ·

N:: 3

Find · partial sum

Steps

Keep $n = 1$ , 3: $(4/\pi )(\sin (\pi /2) + \sin (3\pi /2)/3)$
$= (4/\pi )(1 - 1/3) = (4/\pi )(2/3) \approx 0.85$
More terms push the value toward 1 with Gibbs ripple

Result ·

(4/\pi )(1 - 1/3\cdot (-1))

≈ ...

\approx 1.13

Fourier Transform→Euler's Formula→Convolution Theorem→Wave Equation→