HW #09
Due: 11/13/2023
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    1.

      (a) Obtain the Fourier transform of
      f(x)=⎧
      ⎪
      ⎨
      ⎪
      ⎩
      −1,
      −1 < x < 0
      1,
      0 < x < 1
      0,
      otherwise

      (b) Using the Parseval identity, evaluate
      ⌠
      ⌡       		∞

      −∞ 
      		 ⎛
      ⎝       		1−cos x
      x
      		⎞
      ⎠       		2

       
      		  dx.
      No other method is accepted.



    2. The Parseval identity of the Fourier sine series1 is expressed as
    ⌠
    ⌡     		π

    −π 
    		 {f(x)}2 dx = π ∞
    ∑
    m=1 
    		 bm2,
    where f(x) is an odd function and bm is the Fourier sine coefficient.

      (1) Obtain the Fourier series of f(x) = x (−π < x < π).
      (2) Applying the Parseval identity to the above, obtain
      1 + 1
      22
      		 + 1
      32
      		 + 1
      42
      		 + …

Footnotes:

1
f(x) = ∞
∑
m=1 
 bm sin m x,     bm = 1
π
 ⌠
⌡ 		π

−π 
 f(x) sin m x dx.


File translated from TEX by TTH, version 4.03.
On 06 Nov 2023, 13:49.