MATH307 December 2010
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q6 (e) • Q6 (f) • Q7 (a) • Q7 (b) • Q7 (c) •
Question 06 (f)
Suppose you expanded the same function ƒ(x) = x as in part (a) except on the interval . What would be the form (i.e., do not compute the coefficients) of the Fourier series valid for this interval? What points on the plane would you plot to produce a frequency-amplitude plot from this new Fourier series? (Give the answer in terms of the coefficients in the new expansion.)
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The Fourier coefficients to the function over an interval are given by
How can we transform these integrals to be over instead?
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If we extend the domain from to , we can write the new Fourier coefficients using the techniques from part (a) as
We could go and directly compute these integrals but we want to relate them to the coefficients from part (a) computed on a domain . To do this, we can use a substitution and let . Then . With this transformation we have and and so putting everything together,
where we have recognized that the new integral is precisely what we computed to get the coefficients on a period . Therefore on the Fourier coefficients double and
Therefore we have that and the points to plot for the frequency amplitude are .
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