Fourier Series Lecture Notes
Introduction
- Discussion on Fourier series, focusing on:
- Calculation of coefficients in a Fourier series
- Convergence of Fourier series
- Example function: Square wave with discontinuity at π
Calculating Fourier Coefficients
- Goal: Approximate a function with a sum of trigonometric terms (sine and cosine)
- Key Integrals: Used to compute coefficients
- Integral of sin(mt) × sin(nt)
- If m = n: Result is π
- If m ≠n: Result is 0
- Integral of cos(mt) × cos(nt)
- Integral of cos(mt) × sin(nt)
Deriving Coefficients
- Multiply both sides of the Fourier expansion by sine or cosine and integrate from 0 to 2Ï€
- b_n Coefficients:
- Derived from multiplying both sides by sin(mt)
- Formula:
[ b_m = \frac{1}{\pi} \int_0^{2\pi} f(t) \sin(mt) , dt ]
- a_n Coefficients:
- Derived from multiplying both sides by cos(mt)
- Formula:
[ a_m = \frac{1}{\pi} \int_0^{2\pi} f(t) \cos(mt) , dt ]
- a_0 Coefficient:
- Derived by multiplying by 1 and integrating
- Formula:
[ a_0 = \frac{1}{\pi} \int_0^{2\pi} f(t) , dt ]
Application Example
- Function: 1 from 0 to π, 0 from π to 2π
- Coefficients calculated by adjusting integration limits
- Result:
- [ a_0 = 1 ]
- [ a_m = 0 ] for all m
- [ b_m = \frac{2}{\pi m} ] for odd m (1, 3, 5,...)
Convergence of Fourier Series
- Fourier Convergence Theorem:
- Assumes f and f' are piecewise continuous on the interval [-L, L]
- Fourier series converges to f(t) where f is continuous
- At discontinuities, converges to midpoint of the jump
Conclusion
- Fourier series provides a way to represent functions, especially useful for functions with discontinuities
- Convergence depends on the continuity of the function and its derivative
- Formulas for a_0, a_m, and b_m are critical for deriving the series
- Illustrative example shows approximation quality
References
- Link to previous introductory video for foundational concepts
This lecture highlights the mathematical elegance and practical utility of Fourier series in approximating functions with trigonometric terms.