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University Math Lab: Systems & Structures FREE
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Fourier Series
Learning Objectives
- Decompose periodic functions
- Compute Fourier coefficients
- Understand convergence
Key Concepts
Fourier Series
f(x) = a₀/2 + Σ[aₙcos(nx) + bₙsin(nx)]
Orthogonality
sin and cos functions are orthogonal on [-π, π]
Theory
**Fourier Series** represents a periodic function as sum of sines and cosines:
f(x) = a₀/2 + Σₙ₌₁^∞ [aₙcos(nπx/L) + bₙsin(nπx/L)]
**Coefficients (period 2L):**
- a₀ = (1/L)∫₋ₗᴸ f(x) dx
- aₙ = (1/L)∫₋ₗᴸ f(x)cos(nπx/L) dx
- bₙ = (1/L)∫₋ₗᴸ f(x)sin(nπx/L) dx
**Even functions:** Only cosine terms (bₙ = 0)
**Odd functions:** Only sine terms (aₙ = 0)
Examples
Fourier series of f(x)=x on [-π,π]
Solution: f(x) = 2Σₙ₌₁^∞ (-1)^(n+1) sin(nx)/n