AMAT 415 — Week 3 problems

P11) Compute the Fourier coefficients

\hat{f}(n)=\displaystyle{\frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int}dt},

where f(t)=0 if -\pi\leq t\leq 0 and f(t)=1 if 0<t\leq \pi.  Use your answer to write down both the complex Fourier series of f(t)

\displaystyle{\sum_{n=-\infty}^\infty \hat{f}(n)e^{int}}

as well the real Fourier series of f(t)

\displaystyle{a_0 + \sum_{n=1}^\infty a_n\cos(nt)+b_n\sin(nt)}.

P12) Same as P11), but with f(t)=\pi+t if -\pi\leq t\leq 0 and f(t)=\pi-t if 0<t\leq \pi.

P13) Same as P11), but with f(t)=|t| where -\pi\leq t\leq \pi. (Hint: You can use your solution to P12) here.  What’s the relation between the functions of P12) and P13)? )

P13) Same as P11), but with f(t)=t^2. Differentiate the Fourier series you just computed term by term and compare with the Fourier series of f(t)=2t. (We computed the Fourier series of the sawtooth wave f(t)=t in class.) Could you have used the Fourier series for the sawtooth wave to deduce that of f(t)=t^2 instead of computing it “from scratch”?

P14) Same as P11), but with f(t)=-\sin(t) if -\pi\leq t\leq 0 and f(t)=\sin(t) if 0<t\leq \pi.

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