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Chapter 7: Q5P (page 377)
Use Parseval鈥檚 theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series using problem 9.6.
By Parseval theorem
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Use a trigonometry formula to write the two terms as a single harmonic. Find the period and amplitude. Compare computer plots of your result and the given problem.
(a) Sketch at least three periods of the graph of the function represented by cosine series for f(x)in Problem 9.
(b) Sketch at least three periods of the graph of the exponential Fourier series of period2 for f(x)in Problem 9.
(c) To what value does the cosine series in (a) coverage at x=0? At x=1? At x=2? At x=-2?
(d) To what value does the exponential series in (b) converge at x=0? At x=1? At? At x=-2.
If a violin string is plucked (pulled aside and let go), it is possible to find a formula f(x, t) for the displacement at time t of any point x of the vibrating string from its equilibrium position. It turns out that in solving this problem we need to expand the function f(x, 0), whose graph is the initial shape of the string, in a Fourier sine series. Find this series if a string of length l is pulled aside a small distance h at its center, as shown.
In Problems 17 to 22 you are given f(x) on an interval, say 0 < x < b. Sketch several periods of the even function fcof period 2b, the odd function fsof period 2b, and the function fpof period b, each of which equals f(x)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.
Given
a) Sketch at least three periods of the graph of the function represented by the sine series for f(x). Without finding any series, answer thefollowing question:
b) To what value does the sine series in (a) converge at ? At ? At ? At ?
c)If the given function is continued with the period 2and then is represented by a complex exponential series , what is the value of ?
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