10Chapter 2Fourier SeriesSolutions to Exercises 2.31.(a) and (b) Sincefis odd, all thean’s are zero andbn=2pp0sinnπpdx=-2nπcosnπpπ0=-2nπ(-1)n-1=0ifnis even,4nπifnis odd.Thus the Fourier series is4π∞k=01(2k+ 1)sin(2k+ 1)πpx. At the points of discon-tinuity, the Fourier series converges to the average value of the function.In thiscase, the average value is 0 (as can be seen from the graph.5.(a) and (b) The function is even. It is also continuous for allx. All thebns are0. Also, by computing the area between the graph offand thex-axis, fromx= 0tox=p, we see thata0= 0. Now, using integration by parts, we obtainan=2pp0-2cp(x-p/2) cosnπpxdx=-4cp2p0u(x-p/2)vcosnπpxdx=-4cp2=0pnπ(x-p/2) sinnπpxpx=0-pnπp0sinnπpxdx=-4cp2p2n2π2cosnπpxpx=0=4cn2π2(1-cosnπ)=0ifnis even,8cn2π2ifnis odd.Thus the Fourier series isf(x) =8cπ2∞k=0cos(2
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