Fourier SeriesAnton DzhamayDepartment of MathematicsThe University of MichiganAnn Arbor, MI 48109wPage: http://www.math.lsa.umich.edu/~adzhamemail: adzham@umich.eduCopyright \251 2004 by Anton DzhamayAll rights reserved

Some packages that we use in this worksheet:restart: with(plots):

In this worksheet we define a number of Maple commands that make it easier to compute the Fourier coefficients and Fourier series for a given function and plot different Fourier polynomials (i.e., finite approximations to Fourier Series). We illustrate how to use these commands (and also the Fourier series themselves) by a number of examples.

assume(n,integer); assume(m,integer);Shorthand notation for basic functionss:=(x,n)->sin(n*Pi*x/L):s(x,n); c:=(x,n)->cos(n*Pi*x/L):c(x,n);Fourier sine coefficients for NiMtJSJmRzYjJSJ4Rw== on the interval NiM3JCIiISUiTEc= B:=proc(expr,var,n) simplify(int(expr*s(var,n),var=0..L)/int(s(var,n)*s(var,n),var=0..L)); end proc:B(f(x),x,n);Fourier cosine coefficients for NiMtJSJmRzYjJSJ4Rw== on the interval NiM3JCIiISUiTEc= (note that the formulas are different for NiMvJSJuRyIiIQ== and NiMyIiIhJSJuRw== )A:=proc(expr,var,n) simplify(int(expr*c(var,n),var=0..L)/int(c(var,n)*c(var,n),var=0..L)); end proc:A(f(x),x,0);A(f(x),x,n);Full Fourier coefficients for NiMtJSJmRzYjJSJ4Rw== on the interval NiM3JCwkJSJMRyEiIkYl Bf:=proc(expr,var,n) simplify(int(expr*s(var,n),var=-L..L)/int(s(var,n)*s(var,n),var=-L..L)); end proc:Bf(f(x),x,n); Af:=proc(expr,var,n) simplify(int(expr*c(var,n),var=-L..L)/int(c(var,n)*c(var,n),var=-L..L)); end proc:Af(f(x),x,0);Af(f(x),x,n);Fourier sine series and Fourier sine polynomial for NiMtJSJmRzYjJSJ4Rw== on the interval NiM3JCIiISUiTEc= (The subtle difference here is that sometimes series (that uses sum) has troubles with division by zero. The polynomial (that uses add) does not have this problem, but on the other hand can not evaluate symbolic sums).FPs:=proc(expr,var,n) add(B(expr,var,m)*s(var,m),m=1..n); end proc: FSs:=proc(expr,var,n) sum(B(expr,var,m)*s(var,m),m=1..n); end proc:FSs(f(x),x,infinity);Fourier cosine series and Fourier cosine polynomial for NiMtJSJmRzYjJSJ4Rw== on the interval NiM3JCIiISUiTEc=FPc:=proc(expr,var,n) A(expr,var,0)+add(A(expr,var,m)*c(var,m),m=1..n); end proc: FSc:=proc(expr,var,n) A(expr,var,0)+sum(A(expr,var,m)*c(var,m),m=1..n); end proc:FSc(f(x),x,infinity);Full Fourier series and full Fourier polynomial for NiMtJSJmRzYjJSJ4Rw== on the interval NiM3JCwkJSJMRyEiIkYlFP:=proc(expr,var,n) Af(expr,var,0)+add(Af(expr,var,m)*c(var,m)+Bf(expr,var,m)*s(var,m),m=1..n); end proc: FS:=proc(expr,var,n) Af(expr,var,0)+sum(Af(expr,var,m)*c(var,m)+Bf(expr,var,m)*s(var,m),m=1..n); end proc:FS(f(x),x,infinity);Odd extension of expr from the interval NiM3JCIiISUiTEc= to the whole real lineoddext:=proc(expr,var,L) local x; x:=var - floor((var+L)/(2*L))*2*L; unapply(signum(x)*unapply(expr,var)(abs(x)),var) end proc: Even extension of expr from the interval NiM3JCIiISUiTEc= to the whole real lineevenext:=proc(expr,var,L) local x; x:=var - floor((var+L)/(2*L))*2*L; unapply(unapply(expr,var)(abs(x)),var) end proc: Periodic extension of expr from the interval NiM3JCwkJSJMRyEiIkYl to the whole real linepext:=proc(expr,var,L) unapply(unapply(expr,var)(var - floor((var+L)/(2*L))*2*L),var) end proc:

A constant function:f:=x->1: f(x);B[n]=B(f(x),x,n); seq(B[n]=B(f(x),x,n),n=1..10); FSs(f(x),x,infinity);FPs(f(x),x,10);A[n]=A(f(x),x,n); seq(A[n]=A(f(x),x,n),n=0..10); FSc(f(x),x,infinity);FPc(f(x),x,10);A[n]=Af(f(x),x,n);B[n]=Bf(f(x),x,n); seq(A[n]=Af(f(x),x,n),n=0..10); seq(B[n]=Bf(f(x),x,n),n=1..10); FS(f(x),x,infinity);FP(f(x),x,10);L:=1;Fourier sine series differs from the function NiMtJSJmRzYjJSJ4Rw==.display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPs(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier sine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);Since NiMiIiI= belongs to the basis of the cosine Fourier family and the full Fourier family, its series are given by just the first term, and so the approximation is exact:display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPc(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier cosine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FP(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);Fourier sine series converges to an odd periodic extension of NiMtJSJmRzYjJSJ4Rw==. plot([f(x),FPs(f(x),x,7),oddext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier sine series with %d terms and its odd periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Fourier cosine series converges to an even periodic extension of NiMtJSJmRzYjJSJ4Rw==.plot([f(x),FPc(f(x),x,7),evenext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier cosine series with %d terms and its even periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Full Fourier series converges to a periodic extension of NiMtJSJmRzYjJSJ4Rw==.plot([f(x),FP(f(x),x,7),pext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier series with %d terms and its periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Clean-up:f:='f':L:='L':

An odd function:f:=x->x: f(x);B[n]=B(f(x),x,n); seq(B[n]=B(f(x),x,n),n=1..10); FSs(f(x),x,infinity);FPs(f(x),x,10);A[n]=A(f(x),x,n); seq(A[n]=A(f(x),x,n),n=0..10); FSc(f(x),x,infinity);FPc(f(x),x,10);Since NiMtJSJmRzYjJSJ4Rw== is odd, its even Fourier coefficients (i.e., NiMmJSJBRzYjJSJuRw==) are zero.A[n]=Af(f(x),x,n);B[n]=Bf(f(x),x,n); seq(A[n]=Af(f(x),x,n),n=0..10); seq(B[n]=Bf(f(x),x,n),n=1..10); FS(f(x),x,infinity);FP(f(x),x,10);L:=1;display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPs(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier sine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPc(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier cosine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FP(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);plot([f(x),FPs(f(x),x,7),oddext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier sine series with %d terms and its odd periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FPc(f(x),x,7),evenext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier cosine series with %d terms and its even periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FP(f(x),x,7),pext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier series with %d terms and its periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Clean-up:f:='f':L:='L':

An even function:f:=x->x^2: f(x);B[n]=B(f(x),x,n); seq(B[n]=B(f(x),x,n),n=1..10); FSs(f(x),x,infinity);FPs(f(x),x,10);A[n]=A(f(x),x,n); seq(A[n]=A(f(x),x,n),n=0..10); FSc(f(x),x,infinity);FPc(f(x),x,10);Since the function is even, its odd coefficients are zero.A[n]=Af(f(x),x,n);B[n]=Bf(f(x),x,n); seq(A[n]=Af(f(x),x,n),n=0..10); seq(B[n]=Bf(f(x),x,n),n=1..10); FS(f(x),x,infinity);FP(f(x),x,10);L:=1;display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPs(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier sine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPc(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier cosine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FP(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);plot([f(x),FPs(f(x),x,7),oddext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier sine series with %d terms and its odd periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FPc(f(x),x,7),evenext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier cosine series with %d terms and its even periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FP(f(x),x,7),pext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier series with %d terms and its periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Clean-up:f:='f':L:='L':

f:=x->x^2-x: f(x);B[n]=B(f(x),x,n); seq(B[n]=B(f(x),x,n),n=1..10); FSs(f(x),x,infinity);FPs(f(x),x,10);A[n]=A(f(x),x,n); seq(A[n]=A(f(x),x,n),n=0..10); FSc(f(x),x,infinity);FPc(f(x),x,10);This function is neither even nor odd, and so all of its Fourier coefficients are non-zero:A[n]=Af(f(x),x,n);B[n]=Bf(f(x),x,n); seq(A[n]=Af(f(x),x,n),n=0..10); seq(B[n]=Bf(f(x),x,n),n=1..10); FS(f(x),x,infinity);FP(f(x),x,10);L:=1;display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPs(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier sine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPc(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier cosine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FP(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);plot([f(x),FPs(f(x),x,7),oddext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier sine series with %d terms and its odd periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FPc(f(x),x,7),evenext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier cosine series with %d terms and its even periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FP(f(x),x,7),pext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier series with %d terms and its periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Clean-up:f:='f':L:='L':

In this example NiMtJSJmRzYjJSJ4Rw==is a basic function of the cosine Fourier family and the full Fourier familyf:=x->cos(Pi*x/L): f(x);B[n]=B(f(x),x,n); seq(B[n]=B(f(x),x,n),n=1..10); FSs(f(x),x,infinity);FPs(f(x),x,10);A[n]=A(f(x),x,n); seq(A[n]=A(f(x),x,n),n=0..10); FSc(f(x),x,infinity);FPc(f(x),x,10);A[n]=Af(f(x),x,n);B[n]=Bf(f(x),x,n); seq(A[n]=Af(f(x),x,n),n=0..10); seq(B[n]=Bf(f(x),x,n),n=1..10); FS(f(x),x,infinity);FP(f(x),x,10);L:=1;display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPs(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier sine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPc(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier cosine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FP(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);plot([f(x),FPs(f(x),x,7),oddext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier sine series with %d terms and its odd periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FPc(f(x),x,7),evenext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier cosine series with %d terms and its even periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FP(f(x),x,7),pext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier series with %d terms and its periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Clean-up:f:='f':L:='L':

assume(L>0);f:=x->piecewise(x<0,0,x<L/2,x,x<L,1,0):f(x);Since in this case it is hard to obtain general formulas, we'll take NiMvJSJMRyIiIg== from the very beginning:L:=1;plot(f(x),x=0..L,color=red,thickness=2,scaling=constrained);B[n]=B(f(x),x,n); seq(B[n]=B(f(x),x,n),n=1..10); FSs(f(x),x,infinity);FPs(f(x),x,10);A[n]=A(f(x),x,n); seq(A[n]=A(f(x),x,n),n=0..10); FSc(f(x),x,infinity);FPc(f(x),x,10);A[n]=Af(f(x),x,n);B[n]=Bf(f(x),x,n); seq(A[n]=Af(f(x),x,n),n=0..10); seq(B[n]=Bf(f(x),x,n),n=1..10); FS(f(x),x,infinity);FP(f(x),x,10);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPs(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier sine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FPc(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier cosine series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);display( [plot(f(x),x=-3*L..3*L,color=red,thickness=2,title=sprintf("Function f(x)=%a and ...",f(x))), seq(plot([f(x),FP(f(x),x,5*i+1)],x=-3*L..3*L,color=[red,blue],thickness=2,numpoints=1000, title=sprintf("... and its Fourier series with %d terms and ...",5*i+1)),i=0..3) ] ,scaling=constrained,view=[-3*L..3*L,-3*L..3*L],insequence=true);plot([f(x),FPs(f(x),x,7),oddext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier sine series with %d terms\n and its odd periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FPc(f(x),x,7),evenext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier cosine series with %d terms\n and its even periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);plot([f(x),FP(f(x),x,7),pext(f(x),x,L)(x)],x=-3*L..3*L,color=[red,blue,violet],thickness=2,numpoints=1000, title=sprintf("%a, its Fourier series with %d terms\n and its periodic extension",f(x),7),scaling=constrained,view=[-3*L..3*L,-3*L..3*L]);Clean-up:f:='f':L:='L':

Richard Haberman, Elementary Applied Partial Differential Equations, 3rd edition, Prentice Hall

"While every effort has been made to validate the solutions in this worksheet, Waterloo Maple Inc. and the contributors are not responsible for any errors contained and are not liable for any damages resulting from the use of this material."