# 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.  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.

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.

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