Chem Differential Eq HW Solutions Fall 2011 6

# Chem Differential Eq HW Solutions Fall 2011 6 - 6 Chapter 2...

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6 Chapter 2 Fourier Series Solutions to Exercises 2.2 1. The graph of the Fourier series is identical to the graph of the function, except at the points of discontinuity where the Fourier series is equal to the average of the function at these points, which is 1 2 . 5. We compute the Fourier coeﬃcients using he Euler formulas. Let us ±rst note that since f ( x )= | x | is an even function on the interval - π<x<π , the product f ( x ) sin nx is an odd function. So b n = 1 π ± π - π odd function ² ³´ µ | x | sin nx dx =0 , because the integral of an odd function over a symmetric interval is 0. For the other coeﬃcients, we have a 0 = 1 2 π ± π - π f ( x ) dx = 1 2 π ± π - π | x | dx == 1 2 π ± 0 - π ( - x ) dx + 1 2 π ± π 0 xdx = 1 π ±
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## This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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