Fourier Series
Stub. Orthogonality and Fourier coefficients for periodic functions, general Fourier series on an arbitrary period, convergence/term-by-term integration and differentiation, sine and cosine (half-range) series, and using Fourier series to force a general periodic oscillation.
Represents a periodic (or periodically extended) function as an infinite sum of sines and cosines, using orthogonality of the trigonometric system to extract each coefficient by projection — exactly analogous to expanding a vector in an orthogonal basis. Covers the general-period formulas, term-by-term convergence/integration/differentiation results, the sine/cosine series that come from odd/even extensions, and the payoff: solving forced oscillation with an arbitrary periodic forcing term by superposing the response to each harmonic. This machinery is exactly what separation of variables needs to match boundary/initial data for the heat, wave, and Laplace equations.
Outline (TODO — flesh out each)
- Intro to Fourier series: orthogonality
- Periodic functions and Fourier coefficients
- Fourier series (period )
- General Fourier series (arbitrary period)
- Convergence, integration, and differentiation of Fourier series
- Sine and cosine series
- Forced oscillation and Fourier series