Fourier Series

Description

Demonstrates how a square wave can be decomposed into and reconstructed from sine waves. Individual harmonic components animate in one by one, and the running partial sum curve updates to show convergence toward the square wave.

---

Phases

#	Phase Name	Duration	Description
1	Setup	5s	Title, axes, show target square wave in white
2	1st Harmonic	7s	Show sin(x) term, animate addition to sum, display formula term
3	3rd Harmonic	7s	Add (4/3π)sin(3x), update sum curve, compare to square wave
4	5th Harmonic	7s	Add (4/5π)sin(5x), convergence clearly improving
5	7th Harmonic	7s	Add (4/7π)sin(7x), Gibbs phenomenon visible at jump
6	Full Formula	8s	Show Σ (4/nπ)sin(nx) formula, mention n=1,3,5,...
7	Summary	5s	Overlay of square wave and partial sum, convergence annotation

---

Layout

+--------------------------------------------------+
|  Title: "Fourier Series"                         |
+--------------------------------------------------+
|                                                  |
|   Axes with square wave      | Formula panel    |
|   (white, dashed)            | (right side)     |
|                              |                  |
|   Individual harmonics:      | f(x) = Σ 4/(nπ) |
|   1st (blue), 3rd (teal),    | · sin(nx)        |
|   5th (yellow), 7th (gold)   |                  |
|                              | Current sum:     |
|   Running sum (orange)       | highlighted      |
|                              | terms            |
+--------------------------------------------------+

Area Descriptions

• Left 65%: Main axes showing individual sine harmonics and their cumulative sum
• Right 35%: Fourier series formula and term-by-term display
• Bottom: Gibbs phenomenon note, convergence description

---

Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

---

Notes

• Square wave with period 2π, amplitude 1
• Fourier coefficients: 4/(nπ) for odd n
• Show individual harmonics as thin curves and cumulative sum as thicker curve
• Axes x_range = [-2π, 2π], y_range = [-1.5, 1.5]