*by Karel Srot, Department of mathematics at Masaryk University, Czech Republic, *

karels@mail.muni.cz, *© 2005 Karel Srot*

NOTE: This worksheet solves some examples using the package **Fourier**. This package provides procedures for computing Fourier series of real functions, drawing plots and animations. Especially animations illustrates the convergence of Fourier series in a comprehensive form. The usage of procedures from package **Fourier** is described in its help file*.*

The recent czech/english version of the package **Fourier **(as well as some exported examples) can be found at www.math.muni.cz/~xsrot/frady. Unfortunately, this website is only in czech at the present.

**> restart;**

Making package Fourier available. If it is not correctly installed, it is necessary to use the worksheet with sources.

**> with(Fourier);**

**examples with brief comments**

Find the Fourier series of the function *f(x)=sgn(cos x) *on the interval and draw the plots of partial sums , and together with periodic extension of function *f*.

**> f:=x->signum(cos(x)):**

**> FSeries:=FSeriesOfFunction(f,-Pi..Pi);**

First we put the graph of periodic extension into the variable. Then we use it as a background when drawing the graphs of partial sums.

**> g1:=PeriodicExtension(f,-Pi..Pi,-7..7):**

**> AnimGraphOfFSeries(FSeries,[1,3,5],-7..7,[g1]);**

**> **

**> **

Find the Fourier series of the function *f(x)=abs(sin x) *on the interval and draw the plots of partial sums , , and together with periodic extension of function *f*.

**> f:=x->abs(sin(x));**

**> FSeries:=FSeriesOfFunction(f,-Pi..Pi);**

In earlier versions of module Fourier was neccessary to solve this problem manually (because of 0 in dominator when *n=1*). Now it is solved correctly.

To draw the plot we use the same technique as in previous example..

**> g1:=PeriodicExtension(f,-Pi..Pi,-Pi..Pi):**

**> AnimGraphOfFSeries(FSeries,[1,3,5,7],-Pi..Pi,[g1]);**

**> **

**> **

Find the FS of the function *f(x)=x sin(x) *a draw tha animation which shows the convergence of the FS.

**> f:=x->x*sin(x);**

**> FSeries:=FSeriesOfFunction(f,-Pi..Pi);**

**> g1:=PeriodicExtension(f,-Pi..Pi,-6..6):**

**> AnimGraphOfFSeries(FSeries,10,-6..6,[g1],start=1,insequence=true, commentpoint=[-3,2]);**

**other examples**

**> f:=x->piecewise(x<0,1/2*x,2*x);**

**> FSeries:=FSeriesOfFunction(f,-Pi..Pi);**

**> g1:=PeriodicExtension(f,-Pi..Pi,-6..6):**

**> AnimGraphOfFSeries(FSeries,[1,3,5],-6..6,[g1]);**

**> **

**> **

**> f:=x->piecewise(x<0,a,b);**

**> FSeries:=FSeriesOfFunction(f,-Pi..Pi);**

**> **

**> **

**> **

**> **

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