4038HW19_Su2010

4038HW19_Su2010 - f ( x ) = X a n cos nx + b n sin nx . a ....

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Homework #19 to Hand In Math 4038 Summer 2010 Print Your Name: S how all work: We can give credit only for what you write! Hand in only your own work—nothing copied! Please do not convert exact answers into decimal approximations. Please write neatly so that the grader can read your work easily. You may print a copy of this Fle and write on it, or use a blank paper with your name at the top. The maximum score possible is 10. Question. The family of functions ± cos nx π , sin nx π ² ² ² n =0 , 1 , 2 , 3 ,... ³ is a complete orthonormal set for the interval [ π,π ]. Let f ( x )= x on that interval. Thus there is an expansion
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Unformatted text preview: f ( x ) = X a n cos nx + b n sin nx . a . (5) Find the value of a n for each n . (Hint: Use the scalar product. What is the integral of an odd function on the given interval?) b . (5) Find the value of b n for each n . (Hint: Use the scalar product.) c . (10 bonus points!) Use the results of (a) and (b) together with the Plancherel Identity to nd the numerical value of X 1 1 n 2 . (The bonus points will be based on the correctness of your calculation not on looking up the answer.) 1...
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