3.2 The Fourier Series
  x  
given f ( x) for
we want to choose numbers a0 , a1,a2 , ..... and b0 , b1, b2 , ... so that

1
f ( x)  a0   (an cos( nx)  bn sin( nx))
2
n 1
1
integrate
 cos (mx)
2
2
integrate

π

-π

π
-π
π
-π


1 
f ( x)dx  a0  dx
2 
cos( mx) f ( x)dx 

1

a



0
  2 n1 (an cos(nx)  bn sin( nx))  cos(mx)dx

-π
 sin (mx)
integrate

π


sin( mx) f ( x)dx 

1

   2 a   (a

0
n 1
n

cos( nx)  bn sin( nx))  sin( mx)dx

How to
find these
coeff
a0 
1


π
-π
f ( x)dx
π
am   cos( mx) f ( x)dx
-π
for m  1, 2, ...
bm 
1

π
-π
sin( mx) f ( x)dx
for m  1, 2, ...
3.2 The Fourier Series
f ( x) for
  x  

1
FS ( x)  a0   (an cos( nx)  bn sin( nx))
2
n 1
Theorem 3.1:
at each x FS(x) converges to ( f(x+) + f(x-) )/2
a0 
Example
0
f ( x)  
1
1


π
-π
f ( x)dx
π
  x  0
0 x 
am   cos( mx) f ( x)dx
-π
for m  1, 2, ...
1 π
bm   sin( mx) f ( x)dx

-π
for m  1, 2, ...
0
f ( x)  
1
Fourier Series
  x  0
0 x 
1  (1  (1)n )
FS ( x)   
sin( nx)
2 n1
n
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
-3
-2
-1
1
2
3
-3
1
2 sin( x )
2 sin( 3 x )


2

3
-2
-1
1
2
3
1
2 sin( x )
2 sin( 3 x )
2 sin( 5 x )
2 sin( 7 x )




2

3
5
7
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
-3
-2
-1
1
2
0.2
3
-3
-2
-1
15terms
25terms
1
2
3
Fourier Series
0    x  0
f ( x)  
1 0  x  
1
1  (1  (1)n )
FS ( x)   
sin( nx)
2 n1
n
0.8
0.6
0.4
1
0.2
0.8
-3
-2
-1
1
2
3
0.6
125terms
0.4
0.2
-3
-2
-1
1
1000terms
MATHEMATICA
Plot[0.5+Sum[ (1-(-1)^n)*Sin[n x]/(n Pi),{n,1,1000}],{x,-Pi,Pi}];
2
3
3.2 The Fourier Series
given f ( x) for
L x L
we want to choose numbers a0 , a1,a2 , ..... and b0 , b1, b2 , ... so that

1
f ( x)  a0   (an cos( nx)  bn sin( nx))
2
n 1
1 L
nx
an   cos(
) f ( x)dx
-L
L
L
for
n  0,1, 2, ...
1 L
nx
bn   sin(
) f ( x)dx for
L -L
L
n  1, 2, ...
3.2 The Fourier Sine Series
Fourier Sine
given f ( x) for
0 x L
nx
f ( x)   bn sin(
)
L
n 1

2 L
nx
bn   sin(
) f ( x)dx for
0
L
L
Fourier Cosine
given f ( x) for
n  1, 2, ...
0 x L

1
nx
f ( x)  a0   an cos(
)
2
L
n 1
2 L
nx
an   cos(1 L ) f n( x)xdx for
) f ( x)dx
L 0an  -LLcos(
L
L
n  0,1, 2, ...
for
n  0,1, 2, ...