Section 25.5
Fourier Cosine Series
If we even-extend a function, we use cosines. This fits Neumann conditions.
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Introduction
This series includes a constant term (), representing the average value.
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The Coefficients
Formula
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Worked Examples
Example 1: f(x) = x
Expand as a cosine series on .
.
.
Non-zero only for odd n.
Example 2: Visualizing
Notice the "even extension" would look like |x| (a V-shape) on .
Example 3: Convergence Speed
Compare coefficients.
Sine series for had decay.
Cosine series for has decay.
Smoother periodic extension faster convergence.
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Practice Quiz
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