Differential Equations › Section H · Boundary value problems and Fourier series › Topic H.3
Periodic and orthogonal functions
Before a function can be written as a sum of sines and cosines, three tools are needed: periodicity, the symmetry of even and odd functions, and orthogonality. Together they turn the hardest part of Fourier series, finding the coefficients, into a single integral each.
In this topic
- Why these three ideas
- Periodic functions
- Even and odd functions
- Integrals over symmetric intervals
- Orthogonal functions
- Worked example: sines on [0, L]
- Worked example: when orthogonality fails
- The three orthogonal families
- Why orthogonality pays off
- Eigenfunctions are orthogonal
- Common mistakes to avoid
- Key takeaways
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