Science:Math Exam Resources/Courses/MATH307/December 2010/Question 06 (f)

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)  •  

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Question 06 (f)
Suppose you expanded the same function ƒ(x) = x as in part (a) except on the interval $0\leq x\leq 2$. 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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Hint
The Fourier coefficients to the function $x$ over an interval $[0,L]$ are given by $c_{n}={\frac {1}{L}}\int _{0}^{L}xe^{-2\pi {}inx/L}{\textrm {d}}x.$ How can we transform these integrals to be over $[0,1]$ instead?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If we extend the domain from $[0,1]$ to $[0,2]$, we can write the new Fourier coefficients $d_{n}$ using the techniques from part (a) as $d_{n}={\frac {1}{2}}\int _{0}^{2}xe^{-2\pi {}inx/2}{\textrm {d}}x.$ We could go and directly compute these integrals but we want to relate them to the coefficients $c_{n}$ from part (a) computed on a domain $[0,1]$. To do this, we can use a substitution and let $u={\frac {x}{2}}$. Then $\displaystyle {}dx=2du$. With this transformation we have $u(0)=0$ and $u(2)=1$ and so putting everything together, $d_{n}=2\int _{0}^{1}ue^{-2\pi {}inu}{\textrm {d}}u=2c_{n}$ where we have recognized that the new integral is precisely what we computed to get the coefficients on a period $[0,1]$. Therefore on $[0,2]$ the Fourier coefficients double and $d_{n}={\begin{cases}2{\frac {1}{2}}=1,\qquad {}n=0\\2{\frac {i}{2\pi {}n}}={\frac {i}{\pi {}n}},\qquad {}n\neq 0\end{cases}}.$ Therefore we have that $\displaystyle {}|d_{n}|=2|c_{n}|$ and the points to plot for the frequency amplitude are $\displaystyle {}(n,2|c_{n}|)$.

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