Fourier Half and Quarter Range Expansions

>	restart:N:=5:A:=0:B:=2:pleft:=-8:pright:=8:with(plots):

Warning, the name changecoords has been redefined

Extensions of functions

>	f(t):=piecewise(t<A,0, t>A and t<B,4-t^2,t>B ,0);plot(f(t),t=-5..5);

  Even Extension of f:

>	ef:=proc(x) if x>=2*A-B and x < A then evalf(subs(t=2*A-x,f(t))) elif x>=A and x<=B then evalf(subs(t=x,f(t))) else 0 fi; end;

>	ef(1);plot(ef,-5..5);

Odd Extension of f:

>	of:=proc(x) if x>=2*A-B and x < A then evalf(subs(t=2*A-x,-f(t))) elif x>=A and x<=B then evalf(subs(t=x,f(t))) else 0 fi; end;

>	plot(of,-10..10);

>

Periodic Extensions

>	Phalf:=B-A:Pquarter:=2*(B-A):P:=(B-A):

>	fperiodic:=proc(x) local y;y:=x; while y < A do y:=y+P;od; while y > B do y:=y-P;od;evalf(subs(t=y,f(t)));end;

>	fpeven:=proc(x) local y;y:=x; while y < 2*A-B do y:=y+2*Phalf;od; while y > B do y:=y-2*Phalf; od; ef(y);end;

>	fpodd:=proc(x) local y;y:=x; while y < 2*A-B do y:=y+2*Phalf;od; while y > B do y:=y-2*Phalf; od; of(y);end;

>	plot(fpeven,-20..10);

>	plot(fpodd,-20..10);

>	plot(fperiodic,-20..10);

Half Range Expansions

>	p:=B-A:pp:=1/2*(B-A):

>	a[0]:=2/p*int(f(t),t=A..B);aa[0]:=1/pp*int(f(t),t=A..B);

>	aa[n]:=1/pp*int(f(t)*cos(n*Pi*t/pp),t=A..B); bb[n]:=1/pp*int(f(t)*sin(n*Pi*t/pp),t=A..B);

>	a[n]:=2/p*int(f(t)*cos(n*Pi*t/p),t=A..B); b[n]:=2/p*int(f(t)*sin(n*Pi*t/p),t=A..B);

>	fs[N](t):=1/2*aa[0]+sum(aa[n]*cos(n*Pi*t/pp)+bb[n]*sin(n*Pi*t/pp),n=1..N);

>	fsine[N](t):=sum(b[n]*sin(n*Pi*t/p),n=1..N);

>	fcosine[N](t):=1/2*a[0]+ sum(a[n]*cos(n*Pi*t/p),n=1..N);

>	FSINE:=plot(fsine[N](t),t=pleft..pright,color=blue):FCOSINE:=plot(fcosine[N](t),t=pleft..pright,color=blue):F:=plot(fs[N](t),t=pleft..pright,color=blue):

>	FEVEN:=plot(fpeven,pleft..pright,color=red):FPERIODIC:=plot(fperiodic,pleft..pright,color=red):

>	ORIGINAL:=plot(f(t),t=pleft..pright,color=yellow,thickness=5,discont=true):

>	FODD:=plot(fpodd,pleft..pright,color=red):

>	display(FSINE,FODD,ORIGINAL);

>

>	display(FCOSINE,FEVEN,ORIGINAL);

>	display(F,FPERIODIC,ORIGINAL);

>