Fourier Series. Period 2π over the interval [ π, π]. > > restart:with(plots): Warning, the name changecoords has been redefined

Transcription

1 > Fourier Series Period 2 over the interval [, ]. > > restart:with(plots): Warning, the name changecoords has been redefined We begin by considering the function > f:=piecewise(x<-pi/3,0,x<pi/3,,0); 0 x < f := x < 0 otherwise over the interval x =.., whose plot is below. > p[0]:=plot(f,x=-pi..pi,thickness=3,discont=true): display(p[0]); We will approximate this function by Fourier polynomials of degrees through 5. > n:=5; n := 5 We first compute the coefficient a 0 = f( x) dx. 2 > a[0]:=(/(2*pi))*int(f,x=-pi..pi);

2 a 0 := 3 Next we use a loop to compute the coefficients a k = f( x ) sin( kx) dx, along with the Fourier polynomials n F n ( x ) = a 0 + a + k cos( kx) k = n k = b k sin( kx ) for n =..5. f( x ) cos( kx) dx and b k = > for k from to n do a[k]:=(/pi)*int(f*cos(k*x),x=-pi..pi); b[k]:=(/pi)*int(f*sin(k*x),x=-pi..pi); F[k]:=a[0]+sum('a[i]*cos(i*x)','i'=..k)+sum('b[i]*sin(i*x)','i'=..k ); od; 3 a := b := 0 3 cos( x ) F := + 3 a 2 := 2 b 2 := 0 3 cos( x ) 2 3 cos( 2 x) F 2 := + + a 3 := 0 b 3 := 0 3 cos( x ) 2 3 cos( 2 x) F 3 := + + a 4 := 3 4 b 4 := 0 3 cos( x ) 2 3 cos( 2 x ) F 4 := + + a 5 := 3 5 b 5 := cos( 4 x)

3 3 cos( x ) 2 3 cos( 2 x ) 3 cos( 4 x) F 5 := a 6 := 0 b 6 := 0 3 cos( x ) 2 3 cos( 2 x ) 3 cos( 4 x) F 6 := a 7 := 7 b 7 := 0 3 cos( x) 2 3 cos( 2 x ) 3 cos( 4 x ) 3 cos( 5 x) F 7 := a 8 := 8 b 8 := 0 3 cos( x ) 2 3 cos( 2 x ) 3 cos( 4 x) 3 cos( 5 x) F 8 := cos( 8 x ) + a 9 := 0 b 9 := 0 3 cos( x ) 2 3 cos( 2 x ) 3 cos( 4 x) 3 cos( 5 x) F 9 := cos( 8 x ) + a 0 := 0 b 0 := 0 3 cos( x) 2 3 cos( 2 x ) 3 cos( 4 x ) 3 cos( 5 x) F 0 := cos( 5 x) 3 cos( 5 x) 7 3 cos( 7 x) 7 3 cos( 7 x ) 7 3 cos( 7 x ) 7 3 cos( 7 x)

4 8 3 cos( 8 x ) cos( 0 x ) a := b := 0 3 cos( x) 2 3 cos( 2 x ) 3 cos( 4 x ) 3 cos( 5 x) F := cos( 8 x ) 3 cos( 0 x ) 3 cos( x) + 0 a 2 := 0 b 2 := 0 3 cos( x) 2 3 cos( 2 x ) 3 cos( 4 x ) 3 cos( 5 x) F 2 := cos( 8 x ) 3 cos( 0 x ) 3 cos( x) a 3 := b 3 := 0 3 cos( x) 2 3 cos( 2 x ) 3 cos( 4 x ) 3 cos( 5 x) F 3 := cos( 8 x ) 3 cos( 0 x ) 3 cos( x) 3 3 cos ( 3 x ) a 4 := 4 b 4 := 0 3 cos( x) 2 3 cos( 2 x ) 3 cos( 4 x ) 3 cos( 5 x) F 4 := cos( 8 x ) 3 cos( 0 x ) 3 cos( x) 3 3 cos ( 3 x ) cos( 7 x) 7 3 cos( 7 x) 7 3 cos( 7 x) 7 3 cos( 7 x)

5 4 3 cos ( 4 x ) + a 5 := 0 b 5 := 0 3 cos( x) 2 3 cos( 2 x ) 3 cos( 4 x ) 3 cos( 5 x) 7 3 cos( 7 x) F 5 := cos( 8 x ) 3 cos( 0 x ) 3 cos( x) 3 3 cos ( 3 x ) cos ( 4 x ) + Next we plot f( x ) along with each of the F n ( x ) for n =..5. > for k from to n do p[k]:=plot(f[k],x=-pi..pi,thickness=3,discont=true,color=blue): od: > for k from to n do k; display(p[0],p[k]): od;

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12 5 We see that the successive Fourier polynomials give better and better approximations to ( ) f x. Although they may not be as accurate as Taylor polynomials near a given point, they do a better job of approximating a function over an entire interval. They are especially good for approximating

13 periodic functions. Let's extend f( x ) by a period in each direction with x = -... > f:=piecewise(x<-7*pi/3,0,x<-5*pi/3,,x<-pi/3,0,x<pi/3,,x<5*pi/3,0,x< 7*Pi/3,,0); 0 x < 7 x < 5 0 x < f := x < 5 0 x < 7 x < 0 otherwise We graph the extention. > p[0]:=plot(f,x=-3*pi..3*pi,thickness=3,discont=true): display(p[0]); We now plot the extended f( x ) along with each of the F n ( x ) for n =..5. > for k from to n do p[k]:=plot(f[k],x=-3*pi..3*pi,thickness=3,discont=true,color=blue): od: > for k from to n do k;

14 display(p[0],p[k]): od; 2

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21 > Period b over the interval [ b 2, b 2 ]. > > restart:with(plots): Warning, the name changecoords has been redefined We begin by considering the triangular wave function > f:=piecewise(x<-2,0,x<-,x+2,x<0,-x,x<,x,x<2,2-x,0); 0 x < -2 x + 2 x < - f := x x < 0 x x < 2 x x < 2 0 otherwise over the interval x = with period b = 2, whose plot is below. > p[0]:=plot(f,x=-2..2): display(p[0]);

22 We will approximate this function by Fourier polynomials of degrees through 5. > n:=5; n := 5 We first compute the coefficient a 0 = b > a[0]:=(/2)*int(f,x=-..); b 2 b 2 f( x) dx. a 0 := 2 Next we use a loop to compute the coefficients a k = 2 b b 2 b 2 f( x ) sin 2 kx b n F n ( x ) = a a k cos 2 kx b k = here. b 2 b 2 f( x) dx, along with the Fourier polynomials n k = b k sin 2 kx b cos 2 kx b dx and b k = 2 b for n =..5, keeping in mind that b = 2

23 > for k from to n do a[k]:=int(f*cos(pi*k*x),x=-..); b[k]:=int(f*sin(pi*k*x),x=-..); F[k]:=a[0]+sum('a[i]*cos(Pi*i*x)','i'=..k)+sum('b[i]*sin(Pi*i*x)','i '=..k); od; a := 4 2 b := 0 4 cos( x ) F := 2 2 a 2 := 0 b 2 := 0 4 cos( x ) F 2 := 2 2 a 3 := b 3 := 0 4 cos( x ) 4 F 3 := a 4 := 0 b 4 := 0 4 cos( x ) 4 F 4 := a 5 := b 5 := 0 4 cos( x) 4 cos( x) 4 F 5 := a 6 := 0 cos( x ) cos( x ) b 6 := 0 4 cos( x) 4 cos( x) 4 F 6 := cos( 5 x) 2 cos( 5 x) a 7 := b 7 := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 F 7 := a 8 := 0 2 cos( 7 x) 2

24 b 8 := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 F 8 := cos( 7 x) a 9 := b 9 := 0 4 cos( x ) 4 cos( x) 4 cos( 5 x) 4 cos( 7 x) 4 F 9 := a 0 := 0 b 0 := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 cos( 7 x) 4 F 0 := a := b := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 cos( 7 x) 4 F := cos( x) 2 2 a 2 := 0 b 2 := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 cos( 7 x) 4 F 2 := cos( x) 2 2 a 3 := b 3 := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 cos( 7 x) 4 F 3 := cos( x) 4 cos( x) a 4 := 0 b 4 := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 cos( 7 x) 4 F 4 := cos( x) 4 cos( x) cos( 9 x) 2 cos( 9 x) 2 cos( 9 x) 2 cos( 9 x) 2 cos( 9 x ) 2 cos( 9 x) 2

25 a 5 := b 5 := 0 4 cos( x ) 4 cos( x ) 4 cos( 5 x ) 4 cos( 7 x) 4 F 5 := cos( x) 4 cos( x) 4 cos( 5 x ) Next we plot f( x ) along with each of the F n ( x ) for n =..5. > for k from to n do p[k]:=plot(f[k],x=-2..2,color=blue): od: > for k from to n do k; display(p[0],p[k]): od; cos( 9 x) 2 2

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32 5 Again, we see that the successive Fourier polynomials give better and better approximations to ( ) f x. > >