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Fourier Series

Source lecture(s): PHY621 Lec3

Intuition

Any periodic waveform decomposes into sine and cosine waves at integer multiples of a fundamental frequency.

Formal Definition

For \(f(x)\) period \(2L\): \(f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty[a_n\cos(n\pi x/L)+b_n\sin(n\pi x/L)]\).

Mathematical Formulation

\[a_n=\frac{1}{L}\int_{-L}^L f(x)\cos\frac{n\pi x}{L}\,dx$$ $$b_n=\frac{1}{L}\int_{-L}^L f(x)\sin\frac{n\pi x}{L}\,dx\]

Derivation

Use orthogonality: \(\int_{-L}^L \cos(m\pi x/L)\cos(n\pi x/L)\,dx=L\delta_{mn}\) for \(m,n>0\).

Worked Example

Square wave on \([-L,L]\) has only odd sine coefficients.

Common Mistakes

  • Using sine series for even functions.
  • Forgetting \(a_0/2\) normalization.

Quiz

Q1. What is the Fourier basis for even functions on \([-L,L]\)?

Answer

Cosine terms only.