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# Homework3 - (5 Show the formula(2(b Apply(a to ﬁnd the...

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Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 17–24 October 2007 1. (a) Let f : IR IC be 2 L -periodic, ( L > 0), and Riemann integrable. The corresponding Fourier series are + n = -∞ c n e inπx/L , or 1 2 a 0 + n =1 ( a n cos( nπx/L ) + b n sin( nπx/L )) , (1) Then the statements for 2 π -periodic functions hold, with obvious modifica- tions. In particular, we have: If f can be expanded into series of the form (1), then c n = 1 2 L L - l f ( x ) e - inπx/L dx, n = 0 , ± 1 , ± 2 , . . . , (2) a 0 = 1 L L - L f ( x ) dx, (3) a n = 1 L L - L f ( x ) cos( nπx/L ) dx, (4) b n = 1 L L - L f ( x ) sin( nπx/L ) dx,
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Unformatted text preview: , . . . . (5) Show the formula (2). (b) Apply (a) to ﬁnd the Fourier series of the 2-periodic functions f 1 ( x ) = 1-x , f 2 ( x ) = x 2 , (-1 ≤ x < 1). 2. Prove the following formulas: ( a ) ∞ X k =1 (-1) k 2 k-1 = π 4 ( b ) ∞ X 1 n 2 = π 2 6 Hint : Use the Fourier series expansions of the 2 π-periodic functions f ( x ) =   -1 if-π ≤ x < 1 if 0 ≤ x < π , respectively g ( x ) = x 2 ....
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