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Fourier Series

In this notebook we will see some examples of 2L-periodic extensions of functions. Mathematica can be used to plot them. If f(x) is the function you wish to extend, then plot f[Mod[x,2L,-L]].

Example 1

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Here we plot the original function together with its 2-periodic extension.

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Here is the 3-term Fourier series for f(x).

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We can also plot the error in the truncated trigonometric series representation of f(x).

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Here is the 10-term Fourier series for f(x).

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We should note that where the 2-periodic extension of f(x) has a discontinuity the Fourier Series converges to 0 which is the average of -1 and 1, the right and left hand limits respectively of the periodic extension.

We can again examine the error in the approximation.

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Example 2

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Here is the 3-term Fourier series for f(x).

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We can also plot the error in the truncated trigonometric series representation of f(x).

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Here is the 10-term Fourier series for f(x).

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We can again examine the error in the approximation.

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 Created by Mathematica  (February 15, 2006)