Unit 8 of 10

Fourier Series

  • 4 lessons
  • Parts 221 to 228

A Fourier series writes a periodic function as a sum of sines and cosines. It is the mathematics behind sound, signal processing and the analysis of heat and vibration.

This short unit introduces orthogonality of trigonometric functions, Fourier coefficients, when Fourier series converge, and sine and cosine series on half intervals.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Periodic Functions and Orthogonality

  1. Periodic Functions and Trigonometric Orthogonality

    What periods, angular frequency and harmonics are, how an integral works as an inner product of functions, and why sines and cosines of different frequencies are orthogonal.

    Theory part 221Practice part 222

Fourier Expansions

  1. Fourier Coefficients and Fourier Series

    How to write a periodic function as a sum of sines and cosines, compute its Fourier coefficients for any period, use symmetry to save work and read the coefficients as a spectrum.

    Theory part 223Practice part 224

  2. Convergence of Fourier Series

    When the Fourier series of a piecewise smooth periodic function converges, why it lands on the midpoint of a jump, and why the overshoot near a jump never goes away.

    Theory part 225Practice part 226

  3. Fourier Sine and Cosine Series

    How a function given only on an interval from 0 to L can be written as a series of sines alone or cosines alone, and how the choice affects the endpoints and the speed of convergence.

    Theory part 227Practice part 228