Section 25.3
Periodic & Orthogonal Functions
Just as we use dot products for vectors, we use integrals to check if functions are "perpendicular".
1
Introduction
Fourier series break complex waves into simple sines and cosines. This works because sines and cosines are orthogonal.
2
Periodic Functions
Definition
for all x.
The smallest is the fundamental period.
3
Orthogonality
The inner product for functions on is:
If the integral is 0, they are orthogonal.
4
Worked Examples
Example 1: Basic Check
Are and orthogonal on ?
.
Odd function on symmetric interval .
Yes, they are orthogonal.
Example 2: Sines and Cosines
Check and on .
.
Integral of sine over full periods is 0.
This is the key to Fourier series!
Example 3: Visual
Positive area cancels negative area exactly.
4
Practice Quiz
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