Fourier Analysis

Phase 3: Differential Equations & Transforms

Syllabus Goal

Master representation of functions as series of sines and cosines, and integral transforms.

Fourier Series

Any periodic function $f(x)$ can be expanded as: $$ f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[ a_n \cos(nx) + b_n \sin(nx) \right] $$

Fourier Transform

For non-periodic functions: $$ F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-ikx} dx $$ $$ f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty F(k) e^{ikx} dk $$

Applications

• Signal processing.
• Solving Partial Differential Equations (Heat equation, Wave equation).
• Quantum Mechanics (Position vs Momentum space).

Implementation in Zig

• Fast Fourier Transform (FFT) algorithms.