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# a4 - f Â g x and f/g x are odd Can you say anything in...

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Math 257/316 – Assignment 4 Due: Wednesday, February 3 1. Evaluate the integrals L - L cos nπx L cos mπx L d x, L - L sin nπx L sin mπx L d x, L - L cos nπx L sin mπx L d x, for integers n and m , in each of the cases n = m and n = m . 2. Suppose that the function f ( x ) is 2 L -periodic: f ( x + 2 L ) = f ( x ) . Show that L - L f ( x ) dx = a +2 L a f ( x ) dx for any value of a . 3. Determine the minimal (fundamental) period of a) sec 3 x, b) ln ( 1 + sin 2 x ) , c) sin x 4 + sin x 7 . 4. Find the Fourier series of the following functions: a) f ( x ) = x + 1 , - 1 x < 0 , 1 - x, 0 x 1 , f ( x + 2) , otherwise . b) f ( x ) = - 1 2 ( π - x ) , 0 x < 2 π, f ( x + 2 π ) , otherwise . 5. A function is called even if f ( - x ) = f ( x ) and odd if f ( - x ) = - f ( x ) . Show the following properties: a) If f ( x ) and g ( x ) are even, then so are ( f ± g )( x ) , ( f · g )( x ) and ( f/g )( x ) . b) If f ( x ) and g ( x ) are odd, then so are ( f ± g )( x ) , but ( f · g )( x ) and ( f/g ) are even. c) If f ( x ) is even and g
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Unformatted text preview: ( f Â· g )( x ) and ( f/g )( x ) are odd. Can you say anything in general about ( f Â± g )( x ) ? 6. Find the Fourier series for the function which is 2 Ï€-periodic, with f ( x ) = x for-Ï€ < x < Ï€ . By considering the series at x = Ï€/ 2 , deduce the value of the alternating sum âˆž X k =0 (-1) k 2 k + 1 = 1-1 3 + 1 5-1 7 + Â·Â·Â· 7. Spreadsheet (you do not need to hand in anything for this question) Familiarize yourself with the Excel Tutorial on the website, and read through the webpage â€˜Using Excel spread-sheets to evaluate Fourier seriesâ€™....
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