HW #07
Due: 11/03/2025
1. Parseval's 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) Prove the above identity.
    (2) Obtain the Fourier series of f(x) = x (−π < x < π).
    (3) Applying Parseval's identity to the above, obtain
    1 + 1
    22
    		 + 1
    32
    		 + 1
    42
    		 + …
2. Solve the following integral equation for u(x).

⌠
⌡ 		∞

−∞ 
 u(x − y) u(y) d y = e− x2.
Hint:
F(e−a x2 ) =   ⎛
√
π
a
 
  e−ω2/(4 a).

Footnotes:

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

−π 
 f(x) sin m x dx.


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On 30 Oct 2025, 21:23.