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Fourier Series

Introduction​

Fourier series is a powerful mathematical tool named after Joseph Fourier, a French mathematician and physicist. It allows us to represent periodic functions as a sum of simple sine and cosine functions.

Basic Concept​

The main idea behind Fourier series is that any periodic function f(x) with period 2π can be expressed as an infinite sum of sines and cosines:

f(x) = a₀/2 + Σ(aₙ cos(nx) + bₙ sin(nx))

Where:

  • a₀, aₙ, and bₙ are constants (Fourier coefficients)
  • n is an integer that goes from 1 to infinity

Fourier Coefficients​

The Fourier coefficients are calculated using the following formulas:

  1. a₀ = (1/π) ∫ f(x) dx (from -π to π)
  2. aₙ = (1/π) ∫ f(x) cos(nx) dx (from -π to π)
  3. bₙ = (1/π) ∫ f(x) sin(nx) dx (from -π to π)

These integrals essentially measure how much of each frequency component is present in the original function.

Types of Fourier Series​

  1. Full Fourier Series: Contains both sine and cosine terms.
  2. Sine Series: Contains only sine terms (used for odd functions).
  3. Cosine Series: Contains only cosine terms (used for even functions).

Convergence​

The Fourier series converges to the original function under certain conditions:

  • At points of continuity, the series converges to the function value.
  • At jump discontinuities, the series converges to the average of the left and right limits.

Importance in Mathematics and Physics​

Fourier series is fundamental in:

  • Solving partial differential equations
  • Analyzing periodic phenomena
  • Understanding harmonic oscillations

Example: Square Wave​

A simple example of a Fourier series is the representation of a square wave:

f(x) = (4/π) * (sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + ...)

This series converges to a square wave as more terms are added.

Practical Considerations​

When working with Fourier series:

  • More terms generally lead to better approximation
  • The choice of how many terms to use depends on the desired accuracy and computational resources

Exercises​

  1. Try sketching the first few terms of the square wave Fourier series.
  2. Calculate the Fourier coefficients for a simple periodic function, like f(x) = x on [-π, π].
  3. Explore how the number of terms affects the approximation of a given periodic function.