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[MUG] Fourier series

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[MUG] Fourier series
Author: David Garcia Cervetti    Posted: 13/08/1999 20:03:56 GDT
>> From: David Garcia Cervetti "david" Hi you all, I am trying to create an animation of rational functions using series. I have done it fine with Taylor command and series command. Can somebody tell me how to get the Fourier series of a rational function and plot it? Thanks and best regards, -- David Garcia Cervetti Programmer Vizooal, Inc. 22 w 21st Street, Suite 402 New York, NY 10010 Tel: 212.627.1085 www.vizooal.com Email: "david"

[MUG] Re: Fourier series
Author: Maple Group    Posted: 23/08/1999 19:37:40 GDT
>> From: Maple Group "maple_gr" | I am trying to create an animation of rational functions using series. I | have done it fine with Taylor command and series command. | Can somebody tell me how to get the Fourier series of a rational | function and plot it? -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Fri, 20 Aug 1999 15:02:04 -0500 (CDT) From: Mike May "maymk" To: "maple-list" Subject: Fourier series I did a worksheet for my linear algebra class that does precisely that. (Atually it is part of a series of worksheets. Others deal with orthogonal polynomials.) The files are available via anonymous ftp at euler.slu.edu changing directories to Courseware and then to LinearAlgebra the file OnLine4-3-FourierApprox.mws is what you are looking for. You also can get to the files over the web Go to euler.slu.edu, follow the buttons through Teaching, Courseware, and Linear Algebra to http://euler.slu.edu/courseware/LinearAlgebra/LinAlgMaster.html Mike May "maymk" -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "cassco" Date: Fri, 20 Aug 1999 18:48:39 -0700 To: "maple-list" Subject: Fourier series I compute the coefficients of the Fourier series as follows; Given F(t), a0=(1/T)*int(F,t=0..infinity) an=(1/T)*int(F*sin(n*omega*t),t=0..infinity) bn=(1/T)*int(F*cos(n*omega*t),t=0..infinity) where T= period, and omega=2*Pi/T Tom Casselman "cassco"

[MUG] Fourier series
Author: Chuck Baker    Posted: 23/11/2000 21:43:55 GMT
>> From: Chuck Baker "geogra4" Greetings all, I have developed a Fourier series of the following function with a period of 2pie. 0, -pie < x < -pie/2 f(x) = 1, -pie/2 < x < pie/2 0, pie/2 < x < pie I derived the following answer by hand: F(x) = 1/2 + 2/pie(cos x - 1/3 cos3x + 1/5 cos5x +....) How can I derive the same answer using Maple? Thanks, Chuck

[MUG] Re: Fourier series
Author: Maple Group    Posted: 29/11/2000 14:23:03 GMT
>> From: Maple Group "maple_gr" >> From: Chuck Baker "geogra4" | I have developed a Fourier series of the following function with a period | of 2pie. | | 0, -pie < x < -pie/2 | f(x) = 1, -pie/2 < x < pie/2 | 0, pie/2 < x < pie | | I derived the following answer by hand: | F(x) = 1/2 + 2/pie(cos x - 1/3 cos3x + 1/5 cos5x +....) -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Sun, 26 Nov 2000 17:39:20 +0100 From: Backes "backes" To: "maple-list" Subject: Fourier series The following scheme works nicely for all piecewhise defined functions: > ch:= (a,b)->Heaviside(t-a)*Heaviside(-t+b): > T:=2*Pi; omega:=2*Pi/T; N:=5; T := 2 Pi omega := 1 N := 5 > f:=ch(-Pi/2,Pi/2); f := Heaviside(t + 1/2 Pi) Heaviside(-t + 1/2 Pi) > a:=array(0..0); b:=array(1..N); a := array(0 .. 5, []) b := array(1 .. 5, []) > a[0]:=evalf(2/T*int(f, t=0..T)); > for i from 1 to N do > b[i]:=eval(2/T*int(f*sin(i*omega*t), t=0..T) ); > od; a[0] := .5000000000 1 b[1] := ---- Pi 1 b[2] := ---- Pi 1 b[3] := 1/3 ---- Pi b[4] := 0 1 b[5] := 1/5 ---- Pi Greetings from merry old Europe. Backes -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Wed, 29 Nov 2000 13:24:19 +0100 From: Barsuhn "barsuhn" Subject: Fourier series To: "maple-list" Dear Chuck, use the following piece of Maple code > restart; > f:=x->piecewise(x<-Pi/2,0,x<Pi/2,1,0);# define the function > f(x);# inspect the function f > plot(f,-Pi..Pi,-1..2,axes=boxed,scaling=constrained);# get the plot of f > a:=1/Pi*int(f(x)*cos(n*x),x=-Pi..Pi);a:=unapply(a,n); > b:=2/Pi*int(f(x)*sin(n*x),x=-Pi..Pi);b:=unapply(b,n); > number:=10;#as an example > a(0):=1/Pi*int(f(x),x=-Pi..Pi);# a0 in the remember table > Summe:=a(0)/2+sum(a(k)*cos(k*x)+b(k)*sin(k*x),k=1..number); > plot([f(x),Summe],x=-Pi..Pi,-1..2,axes=boxed,scaling=constrained,color=[red,blue ]);# get a picture of f(x) and its partial Fourier sum > This will work for many other functions (period 2*Pi assumed), that you may define using piecewise. You obtain the exact Fourier coefficients. In more complicated cases, however, you will have to replace a(0), a(k), b(k) by evalf(a(0)) etc. in the assignment to Summe. All the best Jurgen -- ------------------- Prof. Dr. Jurgen Barsuhn Fachhochschule Bielefeld University of Applied Sciences Fachbereich Elektrotechnik und Informationstechnik Wilhelm-Bertelsmann-Str. 10 D-33602 Bielefeld -----------

[MUG] Re: Fourier series
Author: Doc RNDr Pavel Hruka CSc    Posted: 30/11/2000 16:17:58 GMT
>> From: "Doc. RNDr. Pavel Hruka CSc." "hruskap" >> From: Chuck Baker "geogra4" | I have developed a Fourier series of the following function with a period | of 2pie. | | 0, -pie < x < -pie/2 | f(x) = 1, -pie/2 < x < pie/2 | 0, pie/2 < x < pie | | I derived the following answer by hand: | F(x) = 1/2 + 2/pie(cos x - 1/3 cos3x + 1/5 cos5x +....) The following procedure evaluates the Fourier series up to n*x term: > Fourier_series:=proc(func,xrange::name=range,n::posint) > local x,a,b,l,k,j,p,q,partsum; > a:=lhs(rhs(xrange)); > b:=rhs(rhs(xrange)); > l:=b-a; > x:=2*Pi*lhs(xrange)/l; > partsum:=1/l*int(func,xrange); > for k to n do # terms of Fourier series: > partsum:=partsum+2/l*int(func*sin(k*x),xrange)*sin(k*x)+2/l*int(func*cos(k*x),xr ange)*cos(k*x); > end do; > partsum; > end proc: #Your function: > f:= x->piecewise(x<-Pi/2,0,x<Pi/2,1,x<Pi,0): # Result: > Fourier_series(f(x),x=-Pi..Pi,6); 2 cos(x) cos(3 x) 2/5 cos(5 x) 1/2 + -------- - 2/3 -------- + ------------ Pi Pi Pi All the best --------------------------------------------------------- Doc. RNDr. Pavel Hruka, CSc. Tel. 5/4114 3257 Ustav Fyziky FEI VUT Fax. 5/4114 3133 Technicka 8, 616 00 Brno Zazn.5/4114 3391 ---------------------------------------------------------

[MUG] Re: Fourier series
Author: Maple Group    Posted: 11/12/2000 14:51:13 GMT
>> From: Maple Group "maple_gr" -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Bastero de Eleizalde, Carlos" "cbastero" To: "''" "maple-list" Subject: Fourier series Date: Tue, 5 Dec 2000 16:44:13 +0100 Hi, This is my way of doing it > restart: > f:=x->piecewise(-Pi<x and x<-Pi/2,0,-Pi/2<x and x<Pi/2,1,Pi/2<x and x<Pi,0); # your function defined on [-T/2,T/2] > T:=2*Pi; > a:=n->simplify(2/(T)*int(f(x)*cos(2*n*Pi*x/T),x=-T/2..T/2)); > b:=n->simplify(2/(T)*int(f(x)*sin(2*n*Pi*x/T),x=-T/2..T/2)); > for k from 0 to 20 do a(k) od; # to get some values of a(n) > for k from 0 to 20 do a(k) od; # to get some values of b(n) Cheers Carlos -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Willard, Daniel Dr DUSA-OR" "Daniel.Willard" To: "''" "maple-list" Subject: Fourier series Date: Thu, 7 Dec 2000 15:51:38 -0500 Here's an answer to your fourier transform question: > f:=x->Heaviside(x+Pi/2)-Heaviside(x-Pi/2); > with(inttrans); > fourier(f(t),t,w); > simplify(%,exp); > -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Doc. RNDr. Pavel Hruka CSc." "hruskap" To: Maple Group "maple_gr" Date: Wed, 6 Dec 2000 12:45:05 +0100 (CET) Subject: Fourier series | Can the same principle be applied to a problem that is described by: | | -1, -1 < x < 0 | f(x) = | 1, 0 < x < 1 | | Assuming f(x) is continuous periodically? | Chuck, your function can be written as > f2(x):=piecewise(x<0, -1,x < 1,1); { -1 x < 0 f2(x) := { { 1 x < 1 #Fourier series ( to the term 6x - not present in odd functions) is: > Fourier_series(f2(x),x=-1..1,6); sin(x) 4/3 sin(3 x) 4/5 sin(5 x) 4 ------ + ------------ + ------------ Pi Pi Pi #This is the result. The period in the series evaluation is determined by the interval (x=-1..1) , which is the second parameter of the procedure Fourier_series. Of course, f2(x) is NOT periodic, as you can easily verify by plotting it: > plot(f2(x),x=-10..10); More information on functions with discontinuities are in > ?piecewise In my previous e-mail I used a 'procedure definition' of your function, with an arrow. It can be used here, too, with the same result: > f2:=x->piecewise(x<0, -1,x < 1,1); In that e-mail I used a modified procedure for displaying the n-th partial Fourier series, which requires more local parameters to be declared. I realized later, that three of them are not necessary in your problem. Here is a simpler version of the procedure (MAPLE6.01): > Fourier_series:=proc(func,xrange::name=range,n::posint) > local a,b,l,k,t,partsum; > a:=lhs(rhs(xrange)); > b:=rhs(rhs(xrange)); > l:=b-a; > t:=2*Pi*lhs(xrange)/l; > partsum:=1/l*int(func,xrange); > for k to n do > partsum:=partsum+2/l*int(func*sin(k*t),xrange)*sin(k*x)+2/l*int(func > *cos(k*t),xrange)*cos(k*x); end do; partsum; end proc: All the best Pavel | >> From: "Doc. RNDr. Pavel Hruka CSc." "hruskap" | | >> From: Chuck Baker "geogra4" | | I have developed a Fourier series of the following function with a period | | of 2pie. | | | | 0, -pie < x < -pie/2 | | f(x) = 1, -pie/2 < x < pie/2 | | 0, pie/2 < x < pie | | | | I derived the following answer by hand: | | F(x) = 1/2 + 2/pie(cos x - 1/3 cos3x + 1/5 cos5x +....) | | | The following procedure evaluates the Fourier series up to n*x term: | | > Fourier_series:=proc(func,xrange::name=range,n::posint) | > local x,a,b,l,k,j,p,q,partsum; | > a:=lhs(rhs(xrange)); | > b:=rhs(rhs(xrange)); | > l:=b-a; | > x:=2*Pi*lhs(xrange)/l; | > partsum:=1/l*int(func,xrange); | > for k to n do | # terms of Fourier series: | > partsum:=partsum+2/l*int(func*sin(k*x),xrange)*sin(k*x)+2/l*int(func*cos(k*x),xr ange)*cos(k*x); | > end do; | > partsum; | > end proc: | #Your function: | > f:= x->piecewise(x<-Pi/2,0,x<Pi/2,1,x<Pi,0): | # Result: | > Fourier_series(f(x),x=-Pi..Pi,6); | | 2 cos(x) cos(3 x) 2/5 cos(5 x) | 1/2 + -------- - 2/3 -------- + ------------ | Pi Pi Pi | | All the best | +++++++++++++++++++++++++++++++++++++++++++++++++++++++ Pavel Hruska Department of Physics Faculty of Electrical Engineering and Computer Science Brno University of Technology Technicka 8, 616 00 Brno, CZECH REPUBLIC Tel.: +420-5-4114 3257 +420-5-4114 3391 (secretary) Fax : +420-5-4114 3133 +++++++++++++++++++++++++++++++++++++++++++++++++++++++

[MUG] Re: Fourier series
Author: Wilhelm Werner    Posted: 14/12/2000 08:47:31 GMT
>> From: Wilhelm Werner "werner" Hi MUG members. The numerous contributions to the Fourier series question show that many people are interested in that subject. Maybe my contribution to the Maple Application Center "Symbolic Computation of Fourier Series" is worth mentioning in that context. Regards W.Werner ---------------------------------------------------------------------- Wilhelm Werner Phone: +49-(0)7940-1306-96 FH Heilbronn/Standort Kuenzelsau Fax : +49-(0)7940-1306-20 Daimlerstr. 35 D-74653 Kuenzelsau e-mail: "werner" ----------------------------------------------------------------------

[MUG] Fourier series
Author: Anders Ekdahl    Posted: Mon, 04 Feb 2002 15:34:39 +0100
>> From: Anders Ekdahl "andek565" Is there an easy way to get the Fourier series of a periodic function in Maple V? If I have eg f(x)=cos(x)^2 if want expand it to f(x) =1/2+1/2*cos(2x).

[MUG] Re: Fourier series
Author: Maple User Group    Posted: Thu, 14 Feb 2002 09:00:32 -0500
>> From: Maple User Group "maple_gr" | >> From: Anders Ekdahl "andek565" | Is there an easy way to get the Fourier series of a periodic function | in Maple V? If I have eg f(x)=cos(x)^2 if want expand it to f(x) | =1/2+1/2*cos(2x). -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Alec Mihailovs" "mihailovs" To: "maple-list" Subject: Fourier series Date: Fri, 8 Feb 2002 23:40:42 -0500 For trigonometric polynomials, "combine" will do, > combine(cos(x)^2); 1/2 cos(2 x) + 1/2 In general, there is a module, FourierSeries, developed by Wilhelm Werner, see http://www.mapleapps.com/categories/engineering/engmathematics/html/fouriers ymbol.html Best wishes, Alec Mihailovs http://webpages.shepherd.edu/amihailo -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Bastero de Eleizalde, Carlos" "cbastero" To: "''" Subject: Fourier series Date: Sat, 9 Feb 2002 20:04:51 +0100 Hi Anders, Try this one: T:=evalf(2*Pi); > alpha:=0.; > beta:=alpha+T; > F:=x-> cos(x)^2; > for n from 0 to 10 do a||n:=2*int(F(x)*cos(2*n*Pi*x/T), x=alpha..beta)/T od; > for n from 1 to 10 do b||n:=2*int(F(x)*sin(2*n*Pi*x/T), x=alpha..beta)/T od; Yours Carlos Bastero Universidad de Navarra Spain

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