Classroom Tips and Techniques:
Teaching Fourier Series with Maple - Part 4 

Robert J. Lopez 

Emeritus Professor of Mathematics and Maple Fellow 

Maplesoft 

Initializations 

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Once the files for the Fourier package by Karel Srot are in place, the following command will load its code. 

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The Fourier Package by Karel Srot 

In this fourth (and final) article in a series devoted to Maple implementations of Fourier series calculations, we describe the Fourier package announced by Karel Srot in a post to MaplePrimes on August 21, 2006.  Instructions for downloading and  installing the package are available at www.math.muni.cz/~xsrot/frady.  Although the author notes that this site is primarily in Czech, obtaining the code is straightforward.  Installation instructions within the download are in English. 

Although a significant portion of the functionality in this package is available elsewhere, the nuances of the commands make for a different feel to the package.  In addition, there is a Fourier series Maplet that is worth considering.  Recall that in the first article of this series, we detailed how to implement the calculations for Fourier series using just commands built into Maple.  In the second, we detailed the FourierSeries package by Amir Khanshan.  In the third article in this series, we described the FourierSeries package provided to the Maple Application Center by Wilhelm Werner.  

Table of Commands 

Table 1 summarizes the commands available in Karel Srot's Fourier package.  Note that there are help pages for seven of the eight commands listed when the package is loaded via the with(Fourier) command.  The LimitFunction command has no help page and appears to have been superceded by the PeriodicExtension command. 

Command	Principal Argument	Usage
FSeriesOfFunction		Provides formal Fourier series
GetPartialSum	Series	Provides a partial sum of the Fourier series
FSeriesNthTerm	Series	Provides term of the Fourier series
FourierCoeff		For specified , provides a list of the Fourier coefficients
AnimGraphOfFSeries	Series	Provides sequence of partial sums graphed atop
PeriodicExtension		Provides graph of periodic extension of and optionally shows points at if is a point of discontinuity
DeviationOfPartialSums	Series	Calculates and displays values of or of the differences at any specific
Table 1   Commands in Karel Srot's Fourier package.  "Series" is the object returned by the FSeriesOfFunction command, whereas is the function whose Fourier series is computed.

 

Computing a Fourier Series 

A Fourier series for a function is obtained with the FSeriesOfFunction command.  For example, if the function is given by 

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on the interval its Fourier series is given by 

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Notice that the Fourier coefficients are not returned as separate entities, but are embedded in a formal representation of the resulting series. 

A partial sum of the series is obtained with the GetPartialSum command.  For example, the partial sum containing the terms 

 

would be obtained with 

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where and  

A graph of the periodic extension of visible in Figure 1, is obtained with the PeriodicExtension command. 

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Figure 1   Periodic extension of

 

Because at each the Fourier series converges to at discontinuities or at endpoints of intervals where initial and final values do not agree, the graph of the limit function for the Fourier series may not be identical to the graph of the periodic extension.  Indeed, for given above, the limit function for the Fourier series is the function sketched in Figure 2. 

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Figure 2   Limit function for Fourier series of

 

At discontinuities, the series converges to the midpoint of the jump. 

A graph of and a partial sum of its Fourier series can be obtained with the AnimGraphOfFSeries command.  Figure 3 shows a graph of and the partial sum for which  

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Figure 3   In black, graph of and in red, partial sum for which

 

An animation of the convergence of a sequence of partial sums to the limit function appears in Figure 4. 

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Figure 4   Animation of the convergence of partial sums to limit function

 

The command FSeriesNthTerm provides a specific th term of the Fourier series expansion, whereas the FourierCoeff command provides the coefficient of the th term.   

For example, the second term and the corresponding coefficient(s) of the series 

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are obtained with 

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and

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respectively. 

Bessel's Inequality 

 

Although it can be generalized for arbitrary complete orthonormal sets of functions, Bessel's inequality for the Fourier series can be written as 

  

In order to examine how close this inequality comes to being an equality, the Fourier package contains the DeviationOfPartialSums command, which computes  

  

for given partial sums , and plots a bar-graph of the values of as a function of the index .  For example, we would have Figure 5 if the command is applied to the function  

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Figure 5   Values of for

 

The numbers in the list above the graph are the value of .  We can corroborate this claim by computing the directly, via 

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Notice that the DeviationOfPartialSums command returns when given the parameter . 

The difference at a particular value of can also be obtained with the DeviationOfPartialSums command.  In fact, both the values and a bar-graph of the values are available, as we see in Figure 6.  

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Figure 6   The difference for

 

 

That Figure 6 actually produces the differences claimed is verified by the following direct calculation of these differences. 

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Fourier Series Maplet 

While not an integral component of the Fourier package written by Karel Srot, a Fourier series maplet based on the package is available.  Its application is detailed in Figures 7-9.  In Figure 20, we see the result of what is essentially the FSeriesOfFunction command.  The tildes visible in the maplet window appear in Maple 10, but not in Maple 11. 

Figure 7   Fourier series for generated by Maplet

Figure 8 shows a partial sum displayed in the window opened via the "Open window for computing partial sums" button on the left in the main maplet.  

 

Figure 8  Partial sum generated by "Open window for computing partial sums" button

Figure 9 displays the partial sum one frame of an animation that shows the convergence of partial sums to the limit function for the Fourier series corresponding to the function . 

Figure 9  Animation of convergence of partial sums to Visible is

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