Fourier Series Lecture Notes

Introduction

• Discussing Fourier series and how they approximate functions with trigonometric terms (sine and cosine).
• Previous video covered the basics; this video goes deeper.

Key Questions Addressed

1. How to compute the coefficients (a₀, aₙ, bₙ) in a Fourier series.
2. When does a Fourier series converge and provide a good approximation?

Mathematical Identities

• Integrals of Trigonometric Functions:
  • Integral of sin(m·t) * sin(n·t):
    • Equals π if m = n.
    • Equals 0 if m ≠ n.
  • Integral of cos(m·t) * cos(n·t):
    • Equals π if m = n.
    • Equals 0 if m ≠ n.
  • Integral of cos(m·t) * sin(n·t):
    • Always equals 0.*

Computing Fourier Coefficients

• Objective: Find coefficients a₀, aₙ, bₙ such that function f(t) is approximated by the Fourier series.
• Procedure:
  1. Multiply the Fourier series by sin(m·t) and integrate from 0 to 2π.
     • Only the term with the same sine frequency contributes: b_m * π = integral from 0 to 2π of f(t) * sin(m·t) dt.
     • Solve for b_m:
       • b_m = 1/π * integral from 0 to 2π of f(t) * sin(m·t) dt.
  2. Multiply by cos(m·t) and integrate from 0 to 2π.
     • Only the term with the same cosine frequency contributes: a_m * π = integral from 0 to 2π of f(t) * cos(m·t) dt.
     • Solve for a_m:
       • a_m = 1/π * integral from 0 to 2π of f(t) * cos(m·t) dt.
  3. To find a₀, multiply by 1 and integrate.
     • a₀ = 1/π * integral from 0 to 2π of f(t) dt.*

Example Application

• Function: f(t) = 1 from 0 to π, and 0 from π to 2π.
• Find Coefficients:
  • a₀ = 1.
  • aₙ = 0 for n ≥ 1.
  • For odd m, b_m = 2/(π * m).
  • Resulting Fourier series: 1/2 + Σ (2/(π * n)) * sin(mt) for odd n.*

Convergence of Fourier Series

• Fourier Convergence Theorem:
  • If f and f' are piecewise continuous, the series converges to:
    • f(t) where f is continuous.
    • Midpoint of discontinuity where f is discontinuous.
  • Demonstrates relaxed conditions for convergence.

Conclusion

• Fourier series can approximate functions with both continuous and discontinuous points well.
• Series converges to the function itself or the midpoint of discontinuities as more terms are considered.

Important Formulas

• Fourier Coefficients:
  • a₀ = 1/π * integral from 0 to 2π of f(t) dt
  • aₙ = 1/π * integral from 0 to 2π of f(t) * cos(n·t) dt
  • bₙ = 1/π * integral from 0 to 2π of f(t) * sin(n·t) dt*