Fourier Series Overview

Introduction

• Fourier Series: A powerful and beautiful mathematical tool with applications in differential equations, physics, engineering, and more.
• This is an introductory video; future videos will cover applications, theory, and specific examples.

Periodic Functions

• Definition: A function that repeats its values in regular intervals or periods.
• Example: Sine function sin(x) is 2π-periodic
  • If you move any point on the graph over by 2π, it gives the exact same height.
• Square Wave: A function that alternates between 1 and -1, also 2π-periodic.

Approximation of Functions

• Goal: Approximate complex periodic functions using simpler trigonometric functions (sine and cosine).
• Rough Approximation: Using sin(x) to approximate a square wave.
  • Peaks and troughs of sine roughly align with the square wave values.
  • This is a poor approximation.

Improving the Approximation

• Adding additional sine terms with different frequencies and amplitudes:
  • First Addition: sin(3x) / 3
  • Second Addition: sin(5x) / 5
• These adjustments lead to a better approximation of the square wave.

The Big Idea of Fourier Series

• Concept: Approximate a function by summing trigonometric terms of different frequencies.
• Example: Sum of sine terms 4/pi * Σ(sin(nx) / n) for n = 1, 3, 5...
• Continuous Approximation: Adding more terms results in a better approximation.*

Key Phenomena

• The sum of sine terms is continuous, but at discontinuities, it goes through the midpoint (e.g., zero for a square wave at π).
• Gibbs Phenomenon: Overshooting near discontinuities; common in Fourier series.

Utility of Sine Terms

• Periodic Functions: Since sine is periodic, a good approximation in one period (0 to 2π) implies a good approximation elsewhere.
• Applications: Useful in contexts where functions are inherently periodic, like electrical engineering.

Mathematical Representation

• Fourier Series Expansion: Approximate a periodic function f(t) (period 2π) by
  • Sum of cosine terms with coefficients an
  • Sum of sine terms with coefficients bn
  • Form: a0/2 + Σ(an * cos(nt) + bn * sin(nt))
• Example: Square wave approximation involves zero cosine terms and odd sine terms with specific amplitudes.

Generalized Form

• For a function defined on (-L, L), the Fourier series uses a stretching factor π/L within the trigonometric terms.

Upcoming Topics

• How to find an and bn coefficients.
• Convergence of Fourier series and conditions for convergence.
• How Fourier series applies to solving differential equations.

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